Somesectionsaremarkedwithastar.Thisdoesnotindicatetheyarelessimportantormoredifficultthanothersection.Itonlysignifiesthattheyarenotused elsewhereinthetext.Someofthesesectionscontainthemost“interesting”resultsin thistext.Severalofthestarredsectionsinvolveinvestigatingpropertiesofirrational andtranscendentalnumbers.Amongtheseareproofsthat e and π areirrationalas wellastheexplicitconstructionofsometranscendentalnumbers.
6.1Thecurve y = f (x), linesegmentsconnectingthepoint (a, f (a)) tothepoint (x, f (x)) forseveralvaluesof x, andthetangentline y = f (a)+ f (a)(x a) ......................................92
6.2Acurve y = f (x) with f (a)= f (b)= 0,the x axis,and three linesegments indicatingthepoints c inRolle’sTheoremwhere f (x)= 0, i.e.,thepointswherethetangentlineishorizontal .......101
6.3Thegraphiconthe left showsthecurve y = f (x) andtheline y = L(x) connectingtheendpoints (a, f (a)) and (b, f (b)) of thecurve.Thegraphiconthe right shows g(x)= f (x) L(x) andthe x axis.Inthisillustration a = 1, b = 4, and f (x)= x 2x3 15x2 + 33x 16 /2 Hence, L(x)= 2x and g(x)= x 2x3 15x2 + 33x 20 /2 ............................102
6.4Aconvexcurve y = f (x) andoneait’scords ....................111
6.5Aconvexcurve y = f (x) andsomeofit’scords. a Thefigureon the left illustratesLemma 6.8.1.InthenotationofLemma 6.8.1 (x, f (x)) isthepointwherethecordsintersect. b Thefigureonthe right illustratesExercise 6.8.2.InthenotationofExercise 6.8.2 (z, f (z)) isthepointwherethecordsintersect ....................112
6.6Aconvexcurve y = f (x) andsomeofit’scords.Thefigureonthe left illustrates(6.16).Thefigureonthe right illustratesaddinga thirdcord ..................................................113
7.3Insertingoneadditionalpartitionpoint.Thefigureonthe right has oneadditionalpartitionpoint.Clearly,thetwostepfunctionshave thesamesum ...............................................130
7.4The shadedarea illustrates ∑ S ∑ s. ComparetoFigs. 7.1 and 7.2 ..133
7.5Illustrating ∑ S ∑ s =( f (b) f (a)) b a n intheproofofTheorem 7.3.2.Therectangleonthelefthasheight f (b) f (a) andwidth b a n 135
7.6Illustrating ∑ S ∑ s < ε intheproofofTheorem 7.3.3.Since Mi mi < ε b a , therectangleatthebottomhasheight < ε b a . Clearly,ithaswidth b a .....................................136
7.7Illustratingthepositiveandnegativeparts f ± ofafunction f . In thisexample, f = σ isthepseudosinefunction.Inbothgraphs f isthe thincurve.Inthegraphontheleft f + isthe thickerdashed curve andinthegraphontheright f isthe thickerdashescurve ...139
7.8Thefigureonthe left shows f , s, and S. Thefigureon right shows f + , s+ , and S+ . Illustrating s+ isalowerstepfunctionfor f + , S+ isanupperstepfunctionfor f + ,and A+ k a+ k ≤ Ak ak ...........139
9.1Afunction f , thebanddeterminedby f (x) ε < g(x) < f (x)+ ε andafunction g insidethisbound .............................188
9.2Illustratingthat f n doesnotconvergeuniformlyto f inExample 9.2.2.Thefigureshows f , theband f (x) ε < g(x) < f (x)+ ε with ε = 1 3 , andthefunctions f n , n = 1, 2, 3 ..............................................189
10.1The square istheregion {( j , k ) | 0 ≤ j , k ≤ N } correspondingto thesum ∑N j =0 a j ∑N k =0 bk = ∑N j =0 ∑N k =0 a j bk . The boundary betweenthetwo triangles istheline j + k = N , corresponding tothesum cN = ∑ j +k =N a j bk . The lowertriangle isthe region {( j , k ) | 0 ≤ j + k ≤ N } correspondingtothesum ∑N n=0 cn = ∑ j +k ≤N a j bk . The uppertriangle correspondstothesum ∑N j ,k =0 j +k>N a j bk ................................................211
10.2Theperiodicfunction f (t ) ....................................216
Proof. Letrealnumber x and r > 0begiven.Wemustshowthat Br (x) containsat leastonerationalnumber.Suppose x > 0. Let x = d0 .d1 d2 ··· beaninfinitedecimal representationof x. Let N ≥ 1beanintegersuchthat r > 1/10N . Let y := d0 .d1 ··· dN and z := d0 d1 ··· dN 9 = y + 1/10N Then y and z arerationals. Since y isobtainedfrom x = d0 d1 d2 byreplacingthe dk with k > N by0’s and z isobtainedfrom x byreplacingthe dk with k > N by9’swehave
Hence,
Wherethelastinequalityfollowsfromthechoiceof N . Consequently, z isarational in Br (x).
Thecasewhere x < 0issimilar.If x = 0, then x isalreadyarationalin Br (x).
Exercise1.1.7. If r = 0isrationaland x isirrational,then1/x, x/r, rx and r + x are irrational.
Proof. Letrealnumbers x and r > 0begiven.Wewillshowthat Br (x) containsat leastoneirrationalnumber.Let t betheirrationalin(1.1).Then0 < t < 1. Suppose x > 0. Let x = d0 .d1 d2 ··· beaninfinitedecimalrepresentationof x. Let N ≥ 1be anintegersuchthat r > 1/10N . Then
:= d0 .d1 dN + t 10N isanirrationalsuchthat |x y| ≤ 1 10N < r
Thecase x < 0issimilar.If x = 0, then t /10N isanirrationalin Br (x).
1.2AccumulationPoints
Let D beasubsetof C andlet a besomecomplexnumber.Wesay a isan accumulationpoint of D, iftherearepointsin D \{a} arbitrarilycloseto a Hence,apoint a isanaccumulationpointof D, ifgivenanydistance ε > 0, thereisatleastone pointin D \{a} whosedistanceto a islessthan ε Werestatethisas:
Definition1.2.1. Let D beasubsetof C andlet a besomecomplexnumber.We say a isan accumulationpoint of D, ifforall ε > 0, thereisan x ∈ D, suchthat 0 < |x a| < ε. Insymbols,
Apuncturedballisalsocalleda puncturedneighborhood of c Inthesamemanner theopenball Br (a) issometimescalleda neighborhood of a Intermsofballs,wecanrewritethedefinitionofanaccumulationpointas
If A and B aresets,thenthenotation f : A → B isusedtoindicatethat f isdefined onallof A andhasvaluesin B Ifeveryelementof B isavalueof f , then f issaid tobe onto or surjective.SeeSect. A.4 formorebackgroundinformationregarding functions.
Let D beasubsetof C. Letfunction f : D → C andsuppose a ∈ C isanaccumulationpointof D. Let L ∈ C. Wewouldliketoturnthevaguestatement
f (x) iscloseto L, when x iscloseto a intoaprecisemathematicalstatement,thatis,intoastatementthatcanbedescribed usingsettheoryandlogic.Asastepinthisdirectionwereformulatetheprevious vaguestatementas:
wecanarrangethat f (x) isascloseto L, aswewish,
forall x sufficientlycloseto a
Thefirstpart“that f (x) isascloseto L, aswewish”canbemadepreciseasfollows:
where0 < |x a| < δ says x iscloseto a and“sufficiently”isencodedinthedependenceof δ on ε. Wewant a tobeanaccumulationpointof D, becausethatguaranteesthat D ∩ Bδ (a) isnonempty,thatis,0 < |x a| < δ foratleastone x ∈ D. Hencewehavearrivedat:
Definition1.3.1. Let D ⊆ C, a, L ∈ C, and f : D → C If a isanaccumulationpoint of D, wewillsaythat f (x) converges to L as x goesto a provided:Givenany ε > 0, thereisa δ > 0, suchthatforany x ∈ D with0 < |x a| < δ, wehave | f (x) L| < ε
If g : A → B and C ⊆ A, then g(C ) := {g(x) | x ∈ C } isthe image of C Clearly, g(A) ⊆ B Notethat g isonto,if g(A)= B Usingthisimagenotationwecanwrite (1.5)andtherefore(1.4)as
Restating(1.6)inwords:Givenaball B centeredat L, thereisapuncturedball B centeredat a, suchthat f maps B into B. WewillabbreviateEq.(1.4)as lim x
(x)= L
andsaythat L isthe limit of f as x goesto a Usingnotationthatissimilartosaying that f (x) goesto L as x goesto a, wewillalsoabbreviate(1.4),andthereforealso theequivalentformulations(1.5)and(1.6),as
(x) → L, as x → a
andas
)
Model Nothingmovesinthedefinitionofconvergence.Hence,notationusing arrowstodescribeconvergenceissomewhatdeceptive.Herewediscussamore appropriatemodeloflimx→a f (x)= L
Set-upSupposewehaveacanon.Whenthiscanonisfiredatanangle x to thehorizontheprojectilelandsadistance f (x) fromthecanon.Theintended targetisadistance L fromthecanonand a isananglesuchthat f (a)= L In practice,itisnotpossibletosetupthecanoninsuchawaythattheangle x isdeterminedwithabsoluteprecision,i.e.,suchthat x = a. However,weonly needtheprojectiletolandsoclosetothetargetthatthetargetisdestroyed. Wecallthisthetargettolerance.
DiscussionThefunction f (x) encodesalotofinformation,e.g.,howfarthe targetisabove(orbelow)thecanon(ormorecomplicatedinformationabout theterrain),windspeed,atmosphericpressure,etc.Howclosetothetarget
weneedtheprojectiletolanddependsonhowwellthetargetisfortified. Consideringboththeangletothehorizon x andthedirection y inwhichthe canonisfired,leadstoaproblemwhere f (x + iy) isafunctionmappingthe planeintotheplane—intotheplanebecausetheprojectilecanlandleftor rightofthetargetaswellasinfrontoforbehindthetarget. ProblemGivenatarget,setupthecanonsuchthatthetargetisdestroyed. SolutionForanytargettolerance,thereisanangletolerance,suchthat,ifthe angleadjustmentofthecanonfallswithintheangletolerance,thentheprojectilewilllandwithinthetargettolerance.Usingthesymbols a, x, L, and f (x) werestatethisas:Givenanytargettolerance ε > 0, thereisanangletolerance δ > 0, suchthat,if0 < |x a| < δ, then | f (x) L| < ε SeeTable 1.1.
Example1.3.8. Let f : C \{0}→ C bedeterminedby f (x) := 1 x . Then f (x) → 2as x → 1 2 .
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Title: Ida's new shoes
Author: Madeline Leslie
Release date: March 20, 2024 [eBook #73213]
Language: English
Original publication: Boston: Henry A. Young & Co, 1867
*** START OF THE PROJECT GUTENBERG EBOOK IDA'S NEW SHOES ***
Transcriber's note: Unusual and inconsistent spelling is as printed.
AUNT HATTIE'S LIBRARY
Ida's New Shoes.
BY AUNT HATTIE
[MADELINE LESLIE]
AUTHOR OF THE "BROOKSIDE SERIES," ETC.
"He that ruleth his spirit, is better than he that taketh a city."—SOLOMON
BOSTON:
HENRY A. YOUNG & CO. 24 CORNHILL. 1870.
Entered, according to Act of Congress, in the year 1867, by REV. A. R. BAKER,
In the Clerk's Office of the District Court for the District of Massachusetts.
To
NELLIE, ROLAND COTTON, ANNIE, AND FULLER APPLETON,
CHILDREN OF
MY
BELOVED NEPHEW, THE REV. JOHN COTTON SMITH, D.D.,
THESE SMALL VOLUMES ARE AFFECTIONATELY INSCRIBED, WITH THE EARNEST PRAYER THAT THEIR LIVES MAY PROVE THEM TO BE LAMBS IN THE FOLD OF THE GREAT AND GOOD Shepherd of Israel.
CHAPTER I. THE NEW SHOES
CHAPTER II. IDA'S SNARLED HAIR
CHAPTER III. THE ORANGES
CHAPTER IV. VISIT TO GRANDMA
CHAPTER V. THE DANDELIONS
CHAPTER VI. JOSEPH'S PASSION
CHAPTER VII. THE SICK BOY
CHAPTER VIII. THE COUSINS
IDA'S NEW SHOES.
CHAPTER I. THE NEW SHOES.
"LOOK, aunty, see my pretty new shoes!"
Little Ida danced up and down, holding back the skirt of her dress to display the present papa had brought her.
"Very pretty," said Aunt Mary. "I hope my little niece will never let them carry her into mischief."
"No, indeed, aunty! I'm going to be a real good girl now. See how softly I can step in them."
And she went on tiptoe to the door of the bedroom, where her mother was confined with an attack of nervous headache.
Mrs. Kent's head was bandaged tightly with a wet towel, the room being darkened, in the hope she would be able to drop to sleep.
But Ida did not think of this.
"WHAT ARE YOU DOING?"
She wanted her mamma to see how nicely her foot looked in the new boot. She was so quick in her motions, that before Aunt Mary knew what she was about, she had pushed a cricket to the side of the bed, and jumped up, boots and all.
"O Ida! What are you doing?" exclaimed the lady, with a groan. "I was just falling asleep. Jump right off the bed, dear; you jar my head dreadfully."
"See my shoes, first, mamma! Berty tied 'em up for me."
"Ida Kent, come out of that room this minute!" said Aunt Mary. "Your ma is very sick; and you'll make her worse with your noise."
"No, I won't! I'll make her better. I'm going to comb your hair, mamma. You said I might."
Mrs. Kent groaned again. "Do take her away, Mary. I shall die with this dreadful pain. If I could be quiet one hour, I think I should be better."
Aunt Mary took Ida firmly by the hand.
"I won't go! I won't!" screamed the naughty child, at the top of her voice, clinging at the same time to the bedpost.
"Go this minute, Ida," said her mother, holding her throbbing head between her hands. "And Mary, take off
those new shoes. She mustn't wear them till she can behave better."
The naughty girl gave a scream of passion, and was carried out of the room by Aunt Mary, who was obliged to hold her hands to keep her from scratching anywhere she could reach.
Mamma tried to shut her ears, but no, she could hear the loud, angry screams, until Aunt Mary closed the doors in a distant chamber.
Even then, her head throbbed painfully, as she readily imagined the naughty conduct of her little girl.
"What shall I do with her?" she kept saying to herself. "How can I help her conquer her hasty temper? So affectionate one moment, so passionate the next."
Toward night, Mrs. Kent's pain was relieved. She was able to sit up in bed and take a cup of gruel. Nothing could exceed Ida's fond attention. She ran softly to the entry for a shawl, then upstairs for another pillow, and afterwards sat in a chair, her curly head resting on the bed, her forefinger in her mouth, looking as placid and happy as if no cries of passion had ever distorted her features.
Mrs. Kent was greatly refreshed by the gruel. She sat up in bed long enough for Aunt Mary to smooth her tangled hair, and then lay down, saying—
"I feel as if I could sleep."
Presently she put her hand softly on Ida's cheek. "You're a good girl now," she said. "You're mamma's comfort."
Ida caught her mother's hand and kissed it, then went to sucking her finger again.
By and by papa came home, and rocked his little girl in his arms until the tea-bell rang.
"The sky seems remarkably clear to-night," he said to his sister Mary, after a glance at Ida.
"Yes," she said, smiling; "a storm generally clears the sky; and there has been a terrible one to-day."
"Was that what made her mother's head ache so bad?"
"No, she had gone to bed before; but of course the scream's made it worse."
"The thunder you mean. In what direction did the storm arise?"
"In the direction of the s-h-o-e-s." Aunt Mary spelled the word so that Ida need not understand.
"To look at her now, one would think there never could be a storm."
The little girl sat in a high chair by the side of her father, diligently engaged in eating her bread and milk.
"I know what kind of storms you mean," said Berty, laughing.
"Yes, I suppose so," answered his father. "You are quite a man, you know."
"There were never such storms in his day," remarked Aunt Mary, glancing kindly in the face of her favorite nephew.
"Please, aunty, give Ida some cake!" asked the little girl, holding out her plate.
"Yes, dear;" and then added, "I wish her mother could see her now."
"We have a nice young lady at table this evening," remarked Mr. Kent. "I would to have her come here to tea every night."
"I'm going to be good all the time now," said Ida, smiling in her father's face.
CHAPTER II.
IDA'S SNARLED HAIR.
IDA KENT was a very handsome child. Her face was round and fair, her eyes deep-blue, her mouth small and rosy, and her hair rippled and curled all over her head. This was a great affliction to Ida; for, as it hung in ringlets over her neck, it took a long time in the morning to comb out the snarls.
Mrs. Kent, too, used to dread the job; and was always happy when it was completed without arousing Ida's temper. Many a time she had threatened to cut off the curls; but her husband would not consent. He felt very
proud of them, and could not be made to believe that the child would be better without them.
Mamma used to get a bowl of water, a towel to pin around Ida's neck, the comb and brush all ready before she called the little girl; and then she tried to amuse her by telling some funny stories.
A few mornings after Mrs. Kent's sickness, Ida sat in her mother's lap, and all was going on beautifully, when an unlucky snarl caught in the comb, giving it quite a pull.
As quick as thought, the child turned and struck her mother in the face.
"If you do that again, I'll whip you, naughty mamma!" she cried, her face growing very red and angry.
"Stop, Ida! You mustn't talk so; it's wicked," said Mrs. Kent, holding the child's hands. "You know mamma didn't mean to pull; but I shall have to punish you with a rod, if you strike me, or talk so; it is very wicked."
"I won't have my hair done any more!" screamed the naughty girl, kicking with all her might.
"Ida!" called out her father from the next room. "Stop that! Do you know, child, whom you are talking to?"
Berty came in at this minute, and Mrs. Kent took advantage of this opportunity while her attention was engaged to finish the curls.
In a few moments she had forgotten all about her trouble. She came up to kiss papa, her mouth looking as sweet as a ripe cherry, and then went dancing about the room as happy as happy could be.
When her brother had led her to the parlor, Mrs. Kent said with a sigh, "That is the way she acts more than half the time while I am curling her hair. I have tried whipping, and coaxing, and everything I can think of. Her passion grows worse every day."
"I have always hoped she would outgrow it," answered Mr. Kent; "but I see something must be done."
"If you'll tell me what, I'll thank you," murmured the mother in a discouraged tone.
The next week a lady called to see Aunt Mary. Ida was playing quietly with some blocks in the corner of the room, when the visitor caught a glimpse of her.
"Oh, what a darling little girl!" she exclaimed. "Come and see me, my dear."
Ida obeyed, walked slowly across the floor, and glanced shyly in the lady's face from under her curls.
"You little beauty!" repeated the stranger, taking the child in her lap.
"Don't touch the lady's bonnet," said Aunt Mary, seeing that her friend began to look annoyed at having her flowers pulled out of place.
"I will! I will!" And Ida gave a rose-bud a sudden jerk which left it in her hand.
The lady put her upon the floor; and Aunt Mary was so mortified that she caught her up and carried her screaming from the room.
"I am sorry, for your sake, I spoke to the child," apologized the visitor; "but she looked so smiling I never thought of her having such a temper."
"I never saw such a passionate girl," murmured Aunt Mary, looking as if she were going to cry, "I always feel as if an avalanche were hanging over my head. Sometimes she's as sweet and loving as a June rose; and, in a moment, her temper is beyond control."
"Is she well governed?"
"It is difficult to know what to do with one of her disposition. Both my brother and his wife are very solicitous concerning her. I wonder often at my sister's patience. She has tried whipping and shutting her in a room by herself; but nothing seems to have any lasting effect. I really think half the cause of Mrs. Kent's nervous headaches may be from anxiety about Ida. Yesterday she tried a new plan— she led the child into her bedroom, and, without taking the least notice of her screams and struggles, began to pray for herself and her little girl."
"And what effect did this have?"
"Very quieting. My brother feared it was only the novelty, but sister was encouraged to try it again."
"Do you know, my love, who made you?"
"Yes, mamma; God made me and Berty and all the folks."
"And what did he make you for, Ida?"
"I don't know, mamma."
"He made you and all his creatures, to be good, and to do good."
"I'm good now, mamma; I don't kick any." And Ida looked the picture of self-complacence.
"Yes, dear, no one can be better than you, when you try; but I am sorry to say you don't always try. Do you know the good God in the sky looks down upon you then, and is very much displeased?"
"Then, what do you pray to him for?"
Mrs. Kent sighed. "Because," she answered, "I am afraid, if my little girl does not try to correct her naughty temper, that God will punish her. I ask him to forgive you, and help you do right; and then ask him to give me strength to govern you according to his will."
"I won't never be naughty again, mamma; I'm going to be your good little Ida all the time now."
"Dear child, if I could believe that, a heavy burden would be lifted off my heart."
It was scarcely half an hour later that Berty came in from school eating an orange which a companion had given
him.
"I want an orange, mamma," cried Ida, her face flushing.
"Berty will give you a piece of his; dear."
"I would, mamma; but the juice is all sucked out. See Ida, it's nothing but skin."
"I want an orange! I will have an orange!" began the little girl, kicking and screaming with all her strength.
Mrs. Kent put her handkerchief to her face. The disappointment was too bitter. Ida had seemed so penitent, she had really hoped her heart was touched, and would not show such temper again.
"What a naughty girl!" said Berty. "See, you've made mamma cry!"
Ida stopped kicking; stood for a moment irresolute; then she ran and laid her head in her mother's lap.
"I'm sorry, mamma; I don't want an orange now. Take away your handkerchief. See, I'm a good girl."
Mrs. Kent gazed in the child's face. It was beaming with the new delight of having conquered. She took Ida in her arms; pressing her to her heart.
"You have made mamma very happy," she said, kissing her again and again. "Now I know you do mean to try and conquer those wicked passions. Berty, go to the store, darling, and buy the largest orange you call find. Buy two large ones. We must have them cut up for supper; and Ida shall sprinkle sugar on them and help us all herself from the
glass dish. I want papa and Aunt Mary and nurse to know that my dear little girl is determined to be good; that she has begun already to conquer herself."
I wish you could have seen Ida. She danced about the room, kissing mamma ever so many times, and kept running to the window to see whether Berty was coming.
Mrs. Kent looked as happy as her little girl. "Oh, if this will only last!" she said to herself. And then she lifted her heart in prayer to her heavenly Father, beseeching him to put good thoughts into Ida's mind.
When tea was on the table, mamma brought forward the nice dish of orange, and put it before Ida's plate. Then she took the bowl from the tray and said—
"There, love, you may sprinkle on the sugar now. You have been very patient while I cut the oranges."
Ida took the sugar-spoon and began her work. She did not smile. She looked very solemn as if she felt the importance of the trust.
"Why, what is this?" asked papa, coming behind her on tiptoe.
"O papa!" Ida was so excited she sprang almost out of her chair. "I'm going to help the oranges to you and everybody. I've began to be good, and mamma won't have to pray for me any more."
"You have, indeed, arrived at a state of perfection, if that is no longer necessary," replied Mr. Rent, with a smile; "but I want to hear all the particulars of so important a change."
"Mamma will tell you about it. Oh, I'm so good!" and her eyes sparkled with happiness.
"Papa would like best to hear it from his little girl," said mamma, kissing her. "Then we will sit down to tea, and you may help us to your oranges."
The child looked somewhat confused, not knowing exactly where to begin.
"You know," said Berty, "you began to kick and scream because you wanted my orange."
"Yes, papa; and I saw mamma crying behind her handkerchief; and I didn't kick any more. I said right off I'd be good; and then mamma hugged me awfully. Now may I help to the orange?"
"Wait a minute, dear, till papa has thanked the good God for helping his little girl do right."
Papa kissed her with a tear twinkling in his eye. After he had asked God's blessing on the food, Ida began her pleasant task, passing each a few pieces of the fruit and laughing gayly as Aunt Mary, Berty, and papa said, "Thank you, Miss Kent."
