259
names of these figures, which Frank
answering rightly, he asked,
" What sort of triangle is this ?
names of these figures, which Frank
answering rightly, he asked,
" What sort of triangle is this ?
Childrens - Frank
"
While Mary was saying this, Frank
had time to recollect himself.
" My dear Mary," said he, " I was
cross, and you are very good-humoured,
and perhaps you are right too. Now
go to my place and call to me, and I
will stand in yours, and try if I can
hear you. "
Frank could not hear one word that
Mary said: and Frank acknowledged,
that he had been unreasonable. He
perceived, he said, that the wind,
which had been against his voice,
y3
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? 246 FRANK.
while he had been giving his orders,
had prevented his levelling man from
hearing his " lower and lower. "
" My dear," cried Frank, " now I
recollect it is just like the man, who
fell into the coal pit--in the ' Gentle-
man's Magazine. '"
" Man in the coal pit, in the Gentle-
man's Magazine! " said Mary, "What
can you mean? "
" My dear, do not you remember
the sufferings of Lieutenant George
Spearing? the man who went to a
wood to gather some nuts, and fell
into an old coal pit? "
" Oh, I remember," said Mary, " a
hole seventeen yards deep! and he
heard the robin red breast at day break,
singing just over the mouth of his pit.
Poor fellow! "
" Yes," continued Frank, " and he
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? FRANK. 247
heard the horses going to and from the
mill, and human voices. "
" And the ducks and hens distinctly,"
said Mary.
" And he called, and called," said
Frank, " or, as the book says, made the
best use of his voice, but to no manner
of purpose, for the wind was high, and
blew in a line from the mill to the pit;
so that was the reason that he heard
all that was done there distinctly, as I
heard you, Mary; but they could
never hear him; his voice was carried
by the wind the contrary way, as
mine was, my dear, and I beg your
pardon. "
" Think no more of it," said Mary,
" I am glad we did not quarrel
about it. "
" If we had, it would have been all
my fault," said Frank.
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? 248
FRANK.
" But now let us settle how it shall
be for the future," said Mary. " In-
stead of calling in this high wind, why
should not we make signals, as you
told me the engineer and his levelling
man did, when the man was at too
great a distance to hear his voice? "
" Very true, very right,' said Frank;
" how could I be so foolish as not to
think of that! The simplest thing in
the world! But when I am in a pas-
sion I can never think even of the
very thing I want, and that I know
perfectly well when I am not angry. "
" It is so with every body I believe,"
said Mary.
Justly pleased with herself, Mary
was remarkably exact afterwards in
obeying the signals; and Frank, anxi-
ous to make amends for his foolish
passion, was particularly gentle and
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? FRANK.
249
careful not to be the least impatient.
When they went home, Frank told
his mother of their little dispute.
" Now it is all over," said Mary,
" it was very well you thought of
changing places with me, Frank, other-
wise you never could have been so
soon convinced, that I was in the
right. "
" Now it is all over, I was very
foolish," said Frank; " was not I,
mamma. "
His mother could not deny it.
" But, mamma," said Mary, " we
were not quite so foolish as the two
knights, who fought about the gold
and silver shield. "
Frank had never read the story, and
she had the pleasure of reading it to
him. Let those, who have never read
it, read it now, and may those, who
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? 250
FRANK.
have read it before, recollect it the
next time they want it.
" In the days of knight errantry, one
of our good old British princes set up
a statue to the goddess of victory, in a
point where four roads met together.
In her right hand she held a spear,
and her left rested upon a shield: the
outside of this shield was of gold, and
the inside of silver. On the former
was inscribed, in the old British lan-
guage, 'To the goddess ever favour-
able,' and on the other, ' For four
victories obtained successively over the
Piets, and other inhabitants of the
northern islands. '
"It happened one day, that two
knights completely armed, one in black
armour, the other in white, arrived from
opposite parts of the country at this
statue, just about the same time; and
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? FRANK. 251
as neither of them had seen it before,
they stopped to read the inscription,
and observe the excellence of its work-
manship.
" After contemplating it for some
time, ' This golden shield,' says the
black knight--'Golden shield! ' cried
the white knight, who was as strictly
observing the opposite side; ' why, if
I have my eyes, it is silver. ' ' I know
nothing of your eyes,' replied the black
knight; ' but if ever I saw a golden
shield in my life, this is one. ' ' Yes,'
returned the white knight, smiling, ' it
is very probable, indeed, that they
should expose a shield of gold in so
public a place as this: for my part,
I wonder even a silver one is not
too strong a temptation for the devo-
tion of some people, who pass this
way; and it appears, by the date,
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? 252
FRANK.
that this has not been here above three
years. '
"The black knight could not bear
the smile, with which this was deli-
vered, and grew so warm in the dis-
pute, that it soon ended in a challenge ;
they both therefore turned their horses,
and rode back so far as to have suf-
ficient space for their career: then
fixing their spears in their rests, they
flew at each other with the greatest
fury and impetuosity. Their shock
was so rude, and the blow on each
side so effectual, that they both fell to
the ground much wounded and bruised,
and lay there for some time as in a
trance.
"A good druid, who was travelling
that way, found them in this condition.
The druids were the physicians of those
times, as well as the priests. He had
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? FRANK.
253
a sovereign balsam about him, which
he had composed himself, for he was
very skilful in all the plants that grew
in the fields or the forests; he stanched
their blood, applied his balsam to their
wounds, and brought them as it were
from death to life again. As soon as
they were sufficiently recovered, he
began to inquire into the occasion of
their quarrel. ' Why, this man,' cried
the black knight, ' will have it, that
yonder shield is silver. ' ' And he will
have it,' said the other, ' that it is gold;'
and told him all the particulars of the
affair.
"'Ah,' said the druid, with a sigh,
'you are both of you, my brethren, in
the right, and both of you in the
wrong: had either of you given him-
self time to look at the opposite side of
the shield, as well as that which first
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? 254
FRANK.
presented itself to view, all this pas-
sion and bloodshed might have been
avoided. However, there is a very good
lesson to be learned from the evils that
have befallen you on this occasion.
Permit me, therefore, to entreat you
by all our gods, and by the goddess of
victory in particular, never to enter
into any dispute for the future, till you
have fairly considered both sides of the
question. '"
At breakfast, on the day when the
good-natured engineer was expected,
Frank's eyes turned frequently toward
the window; and Mary watched for
him too, for she longed to look through
his wonderful telescope, and to see
men and mountains on their heads.
As to the rest, she cared little about
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? FRANK.
255
taking angles, she did not know what
that meant, or of what use it could be.
" Mary," said Frank, " you would
be more curious about it if you knew
what I know. "
" And what do you know, Frank,
my dear 1" said Mary.
At this question he felt his know-
ledge shrink into a small compass, and
he answered,
"I cannot say that 1 know much;
but, Mary, look out of the window at
that tower at a distance. You see it?
Well! / believe, mind I say / believe,
I do not say that I am sure -- but I
believe that he could, by taking angles,
tell you how high and how broad it is,
without going nearer to it than we are
now; and I think that he could tell
how far off it is from hence, and how
far from that tower to the mountain
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? 256
FRANK.
opposite, or any other place that he
could see at ever so great a distance
with his telescope. "
" My dear Frank, do you believe
this? " said Mary.
" I do, for I was present," persisted
Frank, " when my father asked him
the height and distance of some moun-
tains, as far off as I could see through
the telescope ; and after looking through
his glass, and making some triangles and
calculations, he answered and told exactly
how high they were, and how far distant. "
Mary thought this was impossible;
but she said,
" There are many ways of doing
things, which I do not yet know ; and
this may be possible, though I cannot
conceive how it can be done. "
" We shall see when the good-
natured engineer comes," said Frank.
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? FRANK.
257
His father asked if he remembered
the definitions which he had learned
of an angle, and a right angle, and a
square, and a triangle. He told Frank,
that unless he had perfectly distinct
ideas of these, he would not be able to
understand what he wished to learn
from his good-natured engineer. Frank
took his father's advice, and first he
showed Mary what is meant by an
angle, or a corner; he drew a square
for Mary, and triangles of different
sorts, and he showed her which was a
right-angled triangle: teaching her,
he found, refreshed his own memory.
Mary copied the figures which he had
drawn for her, and then cut out similar
figures in paper, without looking at the
drawings, that she might be quite sure,
that she had a clear recollection of
what she had learned.
z3
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? 258
FRANK.
The engineer arrived, while Frank's
drawings, and the bits of paper, which
Mary had cut into squares and triangles,
were lying on the table.
" I know what you have been doing
here, my little pupil," said he, smiling
at Frank; " you have been preparing
for me. "
" Yes, sir," said Frank, " and I
believe I know them all; ask me any
questions you please. "
"Show me an angle, then," said
the gentleman.
Frank touched the corner of the
square.
The gentleman desired him to
show him each of the angles in the
square and in the triangle; and Frank
did so.
Then, laying the square and the
triangle before Frank, he asked the
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? FRANK.
259
names of these figures, which Frank
answering rightly, he asked,
" What sort of triangle is this ?
Frank answered, " a right-angled tri-
angle. "
"Show me what you mean by a
right angle. "
Frank showed what he meant, first
in the triangle, and afterwards in the
square.
The engineer then took from his
pocket a flat-hinged rule, and asked
Frank if he could with that rule show
him a right angle.
Frank opened the rule, so as to form
with it two sides of a square, and
pointing to the corner where these two
sides met, he said this was a right
angle.
" Here is a pencil: try if you can
. draw a right angle. "
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? 260
FRANK.
Frank drew a horizontal straight
line.
"Now," whispered Mary, "I know
what you are to do next; you will draw
a perpendicular line in the middle
of that, just as if you were going
to draw the wall of a house. Yes,"
said she, as he drew the line, " I knew
that. "
"Hush, little magpie," whispered
Frank's mother.
Frank pointed to the corner where
the perpendicular and horizontal line
joined, and said, that was a right angle.
" Can you show me another right
angle upon this horizontal line? " said
the engineer. " Do you see only one,
or do you see two 1"
" I see two," said Frank; and he
pointed to the corners on the right
hand and on the left hand of the per-
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? FRANK.
261
pendicular line, where it joined the
horizontal line.
The engineer put his hand upon
Frank's head, and said, " Now I am
satisfied that you know what is meant
by an angle, a right angle, and a
triangle.
Mary whispered something to Frank's
mother at this time, who smiled, and
said to the engineer, " Mary is surprised
that you ask Frank so often to show
you an angle in different things. "
"Yes," said Mary, "as if you could
not believe he knew it. "
" I am very careful on these sub-
jects," said the engineer, " for I know
children are sometimes taught very in-
accurately, and then they have such
confused ideas, that it is impossible to
make them understand what is meant.
A young lad was once sent to me to
be turned into a surveyor, who could
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? 262 FRANK.
for some time understand nothing
that I endeavoured to explain to him;
because, though he talked of an angle,
and a right angle, he did not know
clearly what was meant by either; in
short, he mistook a triangle for an
angle. Had he confessed to me his
ignorance at once, I could have cor-
rected his error. "
" Poor boy, he had been ill taught,
I suppose," said Frank.
" You have been well taught, and
ought to be thankful for it," said the
engineer.
" Would you be so good as to come
to this window, sir," interrupted Frank.
" Do you see that tower at a distance?
Could you by taking angles, as you
stand here, find out its breadth and
height, without going to measure any
part of it, sir ? "
" I could," said the engineer.
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? FRANK.
263
" There, Mary! I was right," cried
Frank. " But now, sir, will you be
so very kind as to explain to me how
it is done ? "
" I would be so very kind, if I
could," answered the good-natured en-
gineer ; " but I cannot, I should only
puzzle you. If I were to attempt to
explain it, you could not understand
me.
" Oh, pray ! pray, sir, try ! " said
Mary, " I dare say Frank would un-
derstand you. "
" If you would only try," said
Frank, " I will tell you honestly, after-
wards, if I don't "
" I am sure you would," said the
engineer; " but I tell you beforehand,
that it is impossible. "
Frank looked at his father, hoping
that he knew him better; and that he
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? 264 FRANK.
would say that it was possible. His
father shook his head, answering,
" It is impossible, my dear, till you
have learned a great deal more. "
" Oh, I am very much disappointed,"
said he, " for I expected that I should
have known all these things this morn-
ing. "
" But could you reasonably expect,
my young friend," said the engineer,
" to know in one morning, in one
hour, in one quarter of an hour, what
I have been many mornings, many
days, not to say years in learning? "
" Certainly not," said Frank, laugh-
ing, " that would be rather unrea-
sonable. "
" Then must Frank wait till he is
grown up quite, mamma? " said Mary.
" No, that is not necessary," said his
mother.
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? FRANK.
265
" How old must he be, mamma,
before he can understand them ? "
" How wise must he be, you should
ask, my dear," said his mother; " for
his being able to understand such
things will not depend upon the num-
ber of years he has lived, but upon
what he learns in those years. "
" True, madam, there is Mr. ,
What's his name ? the gentleman who
rode with us the other day, Mr. Rogers,
who has lived more years than I have,
but you saw that he did not understand
these things," said the engineer.
" Nor wish to understand them,"
said Frank: "that did surprise me. "
" And there is the gardener's boy,
Frank," said his father, " who is not
many years older than you are, and
he understands that which you want
to know. "
vol. i. 2 a
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? 266
FRANK.
" Does he indeed ? " said Frank.
M Yes ; now I remember seeing in his
book drawings of triangles and circles,
and I could not guess of what use they
could be. "
" His father said, as you told me, that
he was learning mathematics," said
Mary, " and trig "
" Trigonometry, I suppose," said
the engineeer; " which, translating the
Greek word into English for you, my
little lady, means the measuring of
triangles. "
" Of triangles! " repeated Frank,
taking up one of the paper triangles
which lay upon the table, and looking
at it " Can measuring this have any
thing to do with the measuring that
tower ? "
" Yes; a great deal to do with it,"
answered the engineer. " I cannot ex-
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? FRANK. 267
plain to you how; but I may, without
giving you any false ideas, tell you in
general, that the power we possess of
measuring that tower, and the most
distant objects that can be seen on
earth, and not those only on earth,
but those in the heavens, depends upon
our understanding the properties of a
triangle. "
" If the gardener's boy has learned
trigo-no-me-try," said Mary, " why can-
not Frank? "
" Is there any quick way of learn-
ing it ? " asked Frank.
" No, there is no quick way," said
the engineer.
" You must go regularly through
this," said his father, taking down a
book from the book-case.
" What is it? " cried Frank, seizing
and opening it. " The very thing I
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? 268 FRANK.
saw with the gardener's son, Euclid's
Elements of Geometry. "
" A square is a figure that has
four "
" Oh, we know that," said Mary,
lookiug over his shoulder.
" But how shall I understand these
drawings of circles and triangles ? " said
Frank: " the line a b is equal to the
line c d ; proposition the 1 st, propo-
sition the 2d; and axiom the 1st,
axiom the 2d: almost as hard sound-
ing and difficult as the beginning of
the Latin grammar. "
" Yes," said his father, " in the be-
ginning of all sciences there are dif-
Acuities; a sort of grammar, which
must be learned, before you can get
on to the smooth and pleasant part. "
" But in this book, and in this
science, you will find," said the engi-
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? FRANK. 269
neer, " that each step leads on securely
to another: not one will ever be lost. "
" That is a comfort," said Frank.
" But," said Mary, <<I hope we
may look through the telescope, and
see the men and mountains standing
on their heads. "
The engineer promised that she
should. But he had some business to
do, before he could comply with her
request; and, in the mean time, the
young people were desired to go out.
While Mary went to put on her
bonnet, Frank was left in the hall by
himself. Several of the engineer's
books and instruments, which had been
taken out of his carriage, were lying
on the hall table, and, among others,
one of the telescopes belonging to his
theodolite.
Frank ventured to take up this tele-
2a 3
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? 270
FRANK.
scope, which he ought not to have
touched; he thought, however, that
he could not do it any harm by just
looking through it. He took off the
brass cover at one end, and slid back
the brass slide at the other end, and
looked through it at the tower, and at
some men who were at work in a distant
field.
" What can be the reason," said he
to himself, " that these men seem to
stand on their heads? This telescope
looks as if it were quite the same as
my father's. I wish I could find out
the reason. I should be so glad to
prove that I could understand it,
though they all say I cannot. "
He saw some very slight wires, as
he thought them, behind one of the
glasses ; and as there were none such in
his father's, he fancied that these i had
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? FRANK.
271
something to do with the secret, which
lie longed to discover.
" I know how to unscrew this
glass," said he, "I will not do it the
least harm. "
He unscrewed the glass, and, looking
into the tube, he could scarcely see
what had appeared to him to have been
wires. He put his hand in to feel for
them. There were no wires, there
was nothing that he could feel -- no-
thing !
While Mary was saying this, Frank
had time to recollect himself.
" My dear Mary," said he, " I was
cross, and you are very good-humoured,
and perhaps you are right too. Now
go to my place and call to me, and I
will stand in yours, and try if I can
hear you. "
Frank could not hear one word that
Mary said: and Frank acknowledged,
that he had been unreasonable. He
perceived, he said, that the wind,
which had been against his voice,
y3
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? 246 FRANK.
while he had been giving his orders,
had prevented his levelling man from
hearing his " lower and lower. "
" My dear," cried Frank, " now I
recollect it is just like the man, who
fell into the coal pit--in the ' Gentle-
man's Magazine. '"
" Man in the coal pit, in the Gentle-
man's Magazine! " said Mary, "What
can you mean? "
" My dear, do not you remember
the sufferings of Lieutenant George
Spearing? the man who went to a
wood to gather some nuts, and fell
into an old coal pit? "
" Oh, I remember," said Mary, " a
hole seventeen yards deep! and he
heard the robin red breast at day break,
singing just over the mouth of his pit.
Poor fellow! "
" Yes," continued Frank, " and he
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? FRANK. 247
heard the horses going to and from the
mill, and human voices. "
" And the ducks and hens distinctly,"
said Mary.
" And he called, and called," said
Frank, " or, as the book says, made the
best use of his voice, but to no manner
of purpose, for the wind was high, and
blew in a line from the mill to the pit;
so that was the reason that he heard
all that was done there distinctly, as I
heard you, Mary; but they could
never hear him; his voice was carried
by the wind the contrary way, as
mine was, my dear, and I beg your
pardon. "
" Think no more of it," said Mary,
" I am glad we did not quarrel
about it. "
" If we had, it would have been all
my fault," said Frank.
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FRANK.
" But now let us settle how it shall
be for the future," said Mary. " In-
stead of calling in this high wind, why
should not we make signals, as you
told me the engineer and his levelling
man did, when the man was at too
great a distance to hear his voice? "
" Very true, very right,' said Frank;
" how could I be so foolish as not to
think of that! The simplest thing in
the world! But when I am in a pas-
sion I can never think even of the
very thing I want, and that I know
perfectly well when I am not angry. "
" It is so with every body I believe,"
said Mary.
Justly pleased with herself, Mary
was remarkably exact afterwards in
obeying the signals; and Frank, anxi-
ous to make amends for his foolish
passion, was particularly gentle and
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? FRANK.
249
careful not to be the least impatient.
When they went home, Frank told
his mother of their little dispute.
" Now it is all over," said Mary,
" it was very well you thought of
changing places with me, Frank, other-
wise you never could have been so
soon convinced, that I was in the
right. "
" Now it is all over, I was very
foolish," said Frank; " was not I,
mamma. "
His mother could not deny it.
" But, mamma," said Mary, " we
were not quite so foolish as the two
knights, who fought about the gold
and silver shield. "
Frank had never read the story, and
she had the pleasure of reading it to
him. Let those, who have never read
it, read it now, and may those, who
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? 250
FRANK.
have read it before, recollect it the
next time they want it.
" In the days of knight errantry, one
of our good old British princes set up
a statue to the goddess of victory, in a
point where four roads met together.
In her right hand she held a spear,
and her left rested upon a shield: the
outside of this shield was of gold, and
the inside of silver. On the former
was inscribed, in the old British lan-
guage, 'To the goddess ever favour-
able,' and on the other, ' For four
victories obtained successively over the
Piets, and other inhabitants of the
northern islands. '
"It happened one day, that two
knights completely armed, one in black
armour, the other in white, arrived from
opposite parts of the country at this
statue, just about the same time; and
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? FRANK. 251
as neither of them had seen it before,
they stopped to read the inscription,
and observe the excellence of its work-
manship.
" After contemplating it for some
time, ' This golden shield,' says the
black knight--'Golden shield! ' cried
the white knight, who was as strictly
observing the opposite side; ' why, if
I have my eyes, it is silver. ' ' I know
nothing of your eyes,' replied the black
knight; ' but if ever I saw a golden
shield in my life, this is one. ' ' Yes,'
returned the white knight, smiling, ' it
is very probable, indeed, that they
should expose a shield of gold in so
public a place as this: for my part,
I wonder even a silver one is not
too strong a temptation for the devo-
tion of some people, who pass this
way; and it appears, by the date,
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? 252
FRANK.
that this has not been here above three
years. '
"The black knight could not bear
the smile, with which this was deli-
vered, and grew so warm in the dis-
pute, that it soon ended in a challenge ;
they both therefore turned their horses,
and rode back so far as to have suf-
ficient space for their career: then
fixing their spears in their rests, they
flew at each other with the greatest
fury and impetuosity. Their shock
was so rude, and the blow on each
side so effectual, that they both fell to
the ground much wounded and bruised,
and lay there for some time as in a
trance.
"A good druid, who was travelling
that way, found them in this condition.
The druids were the physicians of those
times, as well as the priests. He had
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? FRANK.
253
a sovereign balsam about him, which
he had composed himself, for he was
very skilful in all the plants that grew
in the fields or the forests; he stanched
their blood, applied his balsam to their
wounds, and brought them as it were
from death to life again. As soon as
they were sufficiently recovered, he
began to inquire into the occasion of
their quarrel. ' Why, this man,' cried
the black knight, ' will have it, that
yonder shield is silver. ' ' And he will
have it,' said the other, ' that it is gold;'
and told him all the particulars of the
affair.
"'Ah,' said the druid, with a sigh,
'you are both of you, my brethren, in
the right, and both of you in the
wrong: had either of you given him-
self time to look at the opposite side of
the shield, as well as that which first
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? 254
FRANK.
presented itself to view, all this pas-
sion and bloodshed might have been
avoided. However, there is a very good
lesson to be learned from the evils that
have befallen you on this occasion.
Permit me, therefore, to entreat you
by all our gods, and by the goddess of
victory in particular, never to enter
into any dispute for the future, till you
have fairly considered both sides of the
question. '"
At breakfast, on the day when the
good-natured engineer was expected,
Frank's eyes turned frequently toward
the window; and Mary watched for
him too, for she longed to look through
his wonderful telescope, and to see
men and mountains on their heads.
As to the rest, she cared little about
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? FRANK.
255
taking angles, she did not know what
that meant, or of what use it could be.
" Mary," said Frank, " you would
be more curious about it if you knew
what I know. "
" And what do you know, Frank,
my dear 1" said Mary.
At this question he felt his know-
ledge shrink into a small compass, and
he answered,
"I cannot say that 1 know much;
but, Mary, look out of the window at
that tower at a distance. You see it?
Well! / believe, mind I say / believe,
I do not say that I am sure -- but I
believe that he could, by taking angles,
tell you how high and how broad it is,
without going nearer to it than we are
now; and I think that he could tell
how far off it is from hence, and how
far from that tower to the mountain
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? 256
FRANK.
opposite, or any other place that he
could see at ever so great a distance
with his telescope. "
" My dear Frank, do you believe
this? " said Mary.
" I do, for I was present," persisted
Frank, " when my father asked him
the height and distance of some moun-
tains, as far off as I could see through
the telescope ; and after looking through
his glass, and making some triangles and
calculations, he answered and told exactly
how high they were, and how far distant. "
Mary thought this was impossible;
but she said,
" There are many ways of doing
things, which I do not yet know ; and
this may be possible, though I cannot
conceive how it can be done. "
" We shall see when the good-
natured engineer comes," said Frank.
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? FRANK.
257
His father asked if he remembered
the definitions which he had learned
of an angle, and a right angle, and a
square, and a triangle. He told Frank,
that unless he had perfectly distinct
ideas of these, he would not be able to
understand what he wished to learn
from his good-natured engineer. Frank
took his father's advice, and first he
showed Mary what is meant by an
angle, or a corner; he drew a square
for Mary, and triangles of different
sorts, and he showed her which was a
right-angled triangle: teaching her,
he found, refreshed his own memory.
Mary copied the figures which he had
drawn for her, and then cut out similar
figures in paper, without looking at the
drawings, that she might be quite sure,
that she had a clear recollection of
what she had learned.
z3
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? 258
FRANK.
The engineer arrived, while Frank's
drawings, and the bits of paper, which
Mary had cut into squares and triangles,
were lying on the table.
" I know what you have been doing
here, my little pupil," said he, smiling
at Frank; " you have been preparing
for me. "
" Yes, sir," said Frank, " and I
believe I know them all; ask me any
questions you please. "
"Show me an angle, then," said
the gentleman.
Frank touched the corner of the
square.
The gentleman desired him to
show him each of the angles in the
square and in the triangle; and Frank
did so.
Then, laying the square and the
triangle before Frank, he asked the
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? FRANK.
259
names of these figures, which Frank
answering rightly, he asked,
" What sort of triangle is this ?
Frank answered, " a right-angled tri-
angle. "
"Show me what you mean by a
right angle. "
Frank showed what he meant, first
in the triangle, and afterwards in the
square.
The engineer then took from his
pocket a flat-hinged rule, and asked
Frank if he could with that rule show
him a right angle.
Frank opened the rule, so as to form
with it two sides of a square, and
pointing to the corner where these two
sides met, he said this was a right
angle.
" Here is a pencil: try if you can
. draw a right angle. "
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? 260
FRANK.
Frank drew a horizontal straight
line.
"Now," whispered Mary, "I know
what you are to do next; you will draw
a perpendicular line in the middle
of that, just as if you were going
to draw the wall of a house. Yes,"
said she, as he drew the line, " I knew
that. "
"Hush, little magpie," whispered
Frank's mother.
Frank pointed to the corner where
the perpendicular and horizontal line
joined, and said, that was a right angle.
" Can you show me another right
angle upon this horizontal line? " said
the engineer. " Do you see only one,
or do you see two 1"
" I see two," said Frank; and he
pointed to the corners on the right
hand and on the left hand of the per-
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? FRANK.
261
pendicular line, where it joined the
horizontal line.
The engineer put his hand upon
Frank's head, and said, " Now I am
satisfied that you know what is meant
by an angle, a right angle, and a
triangle.
Mary whispered something to Frank's
mother at this time, who smiled, and
said to the engineer, " Mary is surprised
that you ask Frank so often to show
you an angle in different things. "
"Yes," said Mary, "as if you could
not believe he knew it. "
" I am very careful on these sub-
jects," said the engineer, " for I know
children are sometimes taught very in-
accurately, and then they have such
confused ideas, that it is impossible to
make them understand what is meant.
A young lad was once sent to me to
be turned into a surveyor, who could
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? 262 FRANK.
for some time understand nothing
that I endeavoured to explain to him;
because, though he talked of an angle,
and a right angle, he did not know
clearly what was meant by either; in
short, he mistook a triangle for an
angle. Had he confessed to me his
ignorance at once, I could have cor-
rected his error. "
" Poor boy, he had been ill taught,
I suppose," said Frank.
" You have been well taught, and
ought to be thankful for it," said the
engineer.
" Would you be so good as to come
to this window, sir," interrupted Frank.
" Do you see that tower at a distance?
Could you by taking angles, as you
stand here, find out its breadth and
height, without going to measure any
part of it, sir ? "
" I could," said the engineer.
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? FRANK.
263
" There, Mary! I was right," cried
Frank. " But now, sir, will you be
so very kind as to explain to me how
it is done ? "
" I would be so very kind, if I
could," answered the good-natured en-
gineer ; " but I cannot, I should only
puzzle you. If I were to attempt to
explain it, you could not understand
me.
" Oh, pray ! pray, sir, try ! " said
Mary, " I dare say Frank would un-
derstand you. "
" If you would only try," said
Frank, " I will tell you honestly, after-
wards, if I don't "
" I am sure you would," said the
engineer; " but I tell you beforehand,
that it is impossible. "
Frank looked at his father, hoping
that he knew him better; and that he
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? 264 FRANK.
would say that it was possible. His
father shook his head, answering,
" It is impossible, my dear, till you
have learned a great deal more. "
" Oh, I am very much disappointed,"
said he, " for I expected that I should
have known all these things this morn-
ing. "
" But could you reasonably expect,
my young friend," said the engineer,
" to know in one morning, in one
hour, in one quarter of an hour, what
I have been many mornings, many
days, not to say years in learning? "
" Certainly not," said Frank, laugh-
ing, " that would be rather unrea-
sonable. "
" Then must Frank wait till he is
grown up quite, mamma? " said Mary.
" No, that is not necessary," said his
mother.
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? FRANK.
265
" How old must he be, mamma,
before he can understand them ? "
" How wise must he be, you should
ask, my dear," said his mother; " for
his being able to understand such
things will not depend upon the num-
ber of years he has lived, but upon
what he learns in those years. "
" True, madam, there is Mr. ,
What's his name ? the gentleman who
rode with us the other day, Mr. Rogers,
who has lived more years than I have,
but you saw that he did not understand
these things," said the engineer.
" Nor wish to understand them,"
said Frank: "that did surprise me. "
" And there is the gardener's boy,
Frank," said his father, " who is not
many years older than you are, and
he understands that which you want
to know. "
vol. i. 2 a
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? 266
FRANK.
" Does he indeed ? " said Frank.
M Yes ; now I remember seeing in his
book drawings of triangles and circles,
and I could not guess of what use they
could be. "
" His father said, as you told me, that
he was learning mathematics," said
Mary, " and trig "
" Trigonometry, I suppose," said
the engineeer; " which, translating the
Greek word into English for you, my
little lady, means the measuring of
triangles. "
" Of triangles! " repeated Frank,
taking up one of the paper triangles
which lay upon the table, and looking
at it " Can measuring this have any
thing to do with the measuring that
tower ? "
" Yes; a great deal to do with it,"
answered the engineer. " I cannot ex-
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? FRANK. 267
plain to you how; but I may, without
giving you any false ideas, tell you in
general, that the power we possess of
measuring that tower, and the most
distant objects that can be seen on
earth, and not those only on earth,
but those in the heavens, depends upon
our understanding the properties of a
triangle. "
" If the gardener's boy has learned
trigo-no-me-try," said Mary, " why can-
not Frank? "
" Is there any quick way of learn-
ing it ? " asked Frank.
" No, there is no quick way," said
the engineer.
" You must go regularly through
this," said his father, taking down a
book from the book-case.
" What is it? " cried Frank, seizing
and opening it. " The very thing I
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? 268 FRANK.
saw with the gardener's son, Euclid's
Elements of Geometry. "
" A square is a figure that has
four "
" Oh, we know that," said Mary,
lookiug over his shoulder.
" But how shall I understand these
drawings of circles and triangles ? " said
Frank: " the line a b is equal to the
line c d ; proposition the 1 st, propo-
sition the 2d; and axiom the 1st,
axiom the 2d: almost as hard sound-
ing and difficult as the beginning of
the Latin grammar. "
" Yes," said his father, " in the be-
ginning of all sciences there are dif-
Acuities; a sort of grammar, which
must be learned, before you can get
on to the smooth and pleasant part. "
" But in this book, and in this
science, you will find," said the engi-
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? FRANK. 269
neer, " that each step leads on securely
to another: not one will ever be lost. "
" That is a comfort," said Frank.
" But," said Mary, <<I hope we
may look through the telescope, and
see the men and mountains standing
on their heads. "
The engineer promised that she
should. But he had some business to
do, before he could comply with her
request; and, in the mean time, the
young people were desired to go out.
While Mary went to put on her
bonnet, Frank was left in the hall by
himself. Several of the engineer's
books and instruments, which had been
taken out of his carriage, were lying
on the hall table, and, among others,
one of the telescopes belonging to his
theodolite.
Frank ventured to take up this tele-
2a 3
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? 270
FRANK.
scope, which he ought not to have
touched; he thought, however, that
he could not do it any harm by just
looking through it. He took off the
brass cover at one end, and slid back
the brass slide at the other end, and
looked through it at the tower, and at
some men who were at work in a distant
field.
" What can be the reason," said he
to himself, " that these men seem to
stand on their heads? This telescope
looks as if it were quite the same as
my father's. I wish I could find out
the reason. I should be so glad to
prove that I could understand it,
though they all say I cannot. "
He saw some very slight wires, as
he thought them, behind one of the
glasses ; and as there were none such in
his father's, he fancied that these i had
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? FRANK.
271
something to do with the secret, which
lie longed to discover.
" I know how to unscrew this
glass," said he, "I will not do it the
least harm. "
He unscrewed the glass, and, looking
into the tube, he could scarcely see
what had appeared to him to have been
wires. He put his hand in to feel for
them. There were no wires, there
was nothing that he could feel -- no-
thing !
