ARITHMETIC RITTER. QA 135 R51 Hitter., L Southern Branch of the University of California Los Angeles Form L-l This book is DUE on the last date stamped below 281 1925 JUL form L-9-15i-8,'24 PEDAGOGICS APPLIED TO- ARITHMETIC. BY PROFESSOR OF MATHEMATICS IN THE STATE NORMAL SCHOOL, CHICO, GAL. "HE THEREFORE THAT IS ABOUT CHILDREN SHOULD WELL STUDY THEIR NATURES AND APTITUDES, AND SEE BY OFTEN TRIALS WHAT TURN THEY EASILY TAKE, AND WHAT BECOMES THEM ; OBSERVE WHAT THEIR STOCK IS, HOW IT MAY BE IMPROVED, AND WHAT IT IS FIT FOR." Locke. STOCKTON, CAL. LEROY S. ATWOOD, PRINTER. 1891. 61233 COPYRIGHT, 1891, BY C. M. HITTER. QA 13S" CONTENTS. Page PREFACE 7 PART I. THE SUPPLY 11 THE BEGINNING OF NUMBER WORK 13 1 TO 10 13 TENS, 10 TO 100 26 10 TO 20 36 20 TO 100 43 100 TO 1000 49 1000 AND ABOVE 53 THE PRIME NUMBER 58 G. C. D. AND L. C. M 61 FRACTIONS 66 FRACTIONS DECIMALS 75 DECIMALS PER CENT 78 TABULATIONS 81 PART II. THE DEMAND 103 PERCENTAGE PROFIT AND Loss 105 BUSINESS DISCOUNT 107 INSURANCE 108 COMMISSION 100 TAXES Ill STOCKS 113 INTEREST 115 BANKING 120 LONGITUDE AND TIME 124 THE METRIC SYSTEM 133 THE PUBLIC LANDS 137 AVERAGE OF ACCOUNTS 144 PROPORTION 147 GENERAL AVERAGE SHIPPING 149 ALLIGATION 150 MENTAL DISCIPLINE 151 RESUME... ..159 PREFACE. The object that I have had in view in the preparation of this work has been the better teaching of that branch of mathematics whose utility no one questions. Among the masses of the American people arithmetic has been regarded of more import- ance than any other study in the educational curriculum. Other studies must be neglected, if necessary, in order that it may be mastered. This state of affairs being admitted, as well as the great evil that would result from the neglect of other most desirable branches of study, it seems to be the duty of those most con- versant with the temper of our people and the time devoted to arithmetic, to put forth every energy to lessen the time, increase the efficiency, and minimize the useless efforts that attach to the teaching and acquiring of this subject. It is certain that no more fertile field is open in which teachers may reap ap- preciation for their untiring energies; for, as Agassiz has said, ' ' On the broad high road of civilization along which men are ever marching, they pass by unnoticed the land marks of intel- lectual progress, unless they chance to have some direct bearing on what is called the practical side of life. ' ' An experience of eighteen years in teaching in country schools, in city grammar and high schools, and in the State Normal School, has enabled me to see and to feel the benefits of good methods and the evils of poor ones. Method is not arbitrary but rational, not inflexible but natural; and he that profits the most by suggestions and by aids is not he that literally follows, but he that digests and assimilates. Rosencranz says: "The peculiarities of the person who is to be educated and, in fact, all the existing circumstances necessitate an adaptation of the universal aims and ends, that cannot be provided for before 8 PREFACE. hand, but must rather test the ready tact of the educator who knows how to take advantage of the existing conditions to fulfill his desired end." Hence I have aimed, not specifically to put in the mouth of the teacher words and formulae which he shall brainlessly and heartlessly utter in the pupils' hearing, but rather general ideas embodied in formulations that shall be suggestive to the receptive mind, that shall enable the true teacher to see his own faults and correct them, and that shall open up new fields for growth. The underlying principle of education is the self activity of the pupil's mind. For the impressment of entirely new ideas the monologue is conceived to be the natural method of teaching; for the enlargement of the thought, the fixing of a concept, and the preparation of the mind for a new principle, the dialogue is believed to be far preferable. The aids are objects in the hands of both the teacher and the pupil; many and varied in the hands of the small child and gradually decreasing in number and variety as the mind of the child becomes the storehouse of faultless concepts drawn from objective percepts. While coun- seling against eccentricity, I would, in the interest of a live school and a progressive teacher commend the following from Tate: "As children love change and novelty, a good teacher will vary his subjects of instruction as well as his methods of instruction accordingly; his judgment must be exercised in selecting those methods which are most suited to the existing conditions of his school." As the work in numbers progresses and when all that is fundamental, the simple number and the fraction (with its varia- tions, decimals and per cent.), have been thoroughly compre- hended by the pupils, the hand of the teacher should rest more and more lightly upon the pupil; hence the pupils are encouraged to make original investigations, under direction of the teacher at first, into such subjects as Commission, Banking, and Taxes, and report for class consideration what they have learned. Harris says: "All teachers must keep in view the standpoint of the pupil, use illustration, and supply necessary steps to make the connection clear to the pupil. The live teacher is careful to avoid being hampered by the limits of any one method, although he finds use for all on occasions. ' ' It is suggested that the teacher in making use of the methods PREFACE. herein outlined keep in mind that a second step should not be taken until the first has been thoroughly made; to that end it will be found necessary to multiply explanations, questions, and exercises, and to vary the same as much as possible. In written or blackboard work, the end sought should be short solutions and clear oral explanations. This end is most readily reached by having the pupils compare and criticise both the solutions and the explanations. This exercise awakens renewed interest, and is always a feature of a well conducted school. The arrangement or order of presentation of subjects, it is thought, will commend itself as being based upon psychological principles. The supply precedes the demand. It is assumed that the teacher aims to be as useful as possible. To be as useful as possible he must be progressive. He must commingle with the world with his eyes and ears open. He must be sociable. He must be high-minded, honest, truthful, and moral in all respects. He must not forget that he is the cynosure of the school and of the community. He must read educational periodicals, and must study psychological and peda- gogical literature, and test its teachings by his experience, and his experience by its teachings. He must explore the vast domain of general literature, for his own growth and that of his school. He must be faultless in the subjects he is to teach, as regards his technical knowledge. To such as are endowed with industry to pursue such a course, this book is sent with pleasure. In their hands the pupils will grow to honorable manhood and true womanhood. State Normal School, 1 C. M. RITTER. Chico, California, August 4th, 1891. ) PART I. THE SUPPLY. "In the teaching of arithmetic we should first carry the pupils through a simple and comprehensive course of calculation, embodying all, or nearly all, the fundamental operations of num- bers, before we attempt to carry them through the so called systematic course of arithmetic, involving long and irksome calculations, intended to give expertness and skill in the manip- ulation of numbers, rather than to awaken and invigorate the intellectual faculties. ' ' Tate . ONE TO TEN. 13 1 TO 1O. ' ' Make your pupil robust and healthy, in order to make him reasonable and wise." Rousseau. THE BEGINNING OF NUMBER WORK. The experience of the best educators of the world seems to point to substantially the same method of developing the idea of number. Educators differ in minor matters of detail, but in the general plan all are agreed. This is because the nature and workings of the mind have been studied with the end in view of determining its powers, its capabilities, and its natural inclina- tions when in a normal state. The result of these studies has been the definite establishment of the fact that in experience, in self-doing, there is produced the greatest increment ol education. Payne says: "Inasmuch as the object can only be attained by the mental action of the learner, by his observing, remembering, etc., it is clear that what he does, not what the teacher does, is the essential part of the process. " It is therefore conceded that the teacher's chief function should be to take advantage of cir- cumstances, to afford means, and to lead and direct, so that the experience, the self-doing, the self-teaching, may be as effective as possible. "To deny this principle is to give a direct sanction to telling and cramming, which are forbidden by the laws of education. To tell the child what he can learn for himself is to neutralize his efforts; consequently, to defeat all the ends of true education." The child's natural methods of learning are objective; there- fore objects are employed as the representatives of ideas. The child naturally learns all he can concerning an object; therefore, 14 THE SUPPLY. we infer that we should lead him to exhaust a subject. Hence, in the teaching of number, objects are employed until the con- cepts of the numbers themselves are correct and ineradicable. The numbers are considered in order, and everything possible is learned concerning- one number before the next in taken up. Nothing is told the pupil except those things which he cannot by any possibility learn for himself under the skillful direction of the teacher. The conversational method seems particularly adapted to the teaching of number. By it the pupils are unconsciously led through all the combinations possible, and with interest intensified rather than abated. "To be interesting, the questions must deal with familiar things, must be varied, and must be simply expressed. It is not to be expected that a lesson of this nature can succeed unless the children feel that the teacher speaks from a full mind, and is quite at ease." The teacher needs to be ever on the alert for weak places. He must constantly check himself in his impetuosity to go forward. He must be master of the idea that to be over-desirous to advance his pupils is as dangerous as to be apathetic. By means of numerous and varied questions, given out to individuals and to the class, made alive by the use of objects in the hands of both the pupil and the teacher, and by means of reviews and repetitions, ever repeated yet never the same, the patient and tactful teacher will seek to unfold in all its completeness that most inscrutable of creations, the human mind. Plato says: "If we could clearly read what is in our own souls, we should find there a correct record of everything proper to be known. ' ' A specimen method of the development of a number (the number three, for instance) is given, not to be slavishly adopted, but to be studied and adapted to the needs and surroundings of the particular school. If the scientific principles upon which this and the following models are based be comprehended, they will be of much vital force as aids to a systematic and thorough presentation of one subject, and a careful preparation for its logically connected succeeding subject. It is hoped, therefore, that these models will be made the subjects of study, and, if com- prehended and approved, that they will be adapted to the necessities of each particular teacher and class. Many times it will be found that fewer or more questions will be necessary to a thorough teaching of a particular point. Different teachers will ONE TO TEN. 15 differ as to the kind or the number of objects, the kind or the number of questions or directions, and the manner of presenting entirely new matter. These differences are quite unimportant, provided the methods of each teacher be based upon psychologi- cal principles and phenomena. The results will be the same. DeGarmo says: "Only one caution needs to be given. The presupposition of brains on the part of the children must always be made; for they come to a thousand conclusions, and take a thousand steps, in thinking, which the teacher need not painfully point out." The numbers one and two need to be treated in the same logical manner and with the same degree of exactness as the number three, though much less time and attention will be required to reach perfect results. Time however is not the essence of the subject. Thoroughness is the desideratum. THE NUMBER THREE. T. (Holding up two blocks.) How many blocks have I? *C. You have two blocks. T. (Holding up the same two blocks in one hand and picking up another block with the other hand.) How many blocks have I now? C. You have two blocks and one block. T. (Putting the three together.) You may bring me just as many pencils. T. You have brought me three pencils. You may now bring me three marbles. T. That is very well done; now how many blocks have I? C. You have three blocks. T. (Holding up three pencils in one hand.) How many pencils have I? C. You have three pencils. T. How many roses have I ? C. You have three roses. T. John, you may bring me three marbles. T. Mary, you may give Susie three splints. *C. Represents "Child," "Children," or "Class," according to circumstances. 16 THE SUPPLY. T. Susie, how many did Mary give you ? S. She gave me three. T. How many marks have I made on the blackboard? C. You have made three marks. T. (Making only two marks this time.) How many marks have I made this time? C. You have made two marks. T. (Making one mark a little at the right of the two.) How many marks have I made this time? C. You have made one mark. T. How many marks did I make both times? C. You made three marks. T. How many are two marks and one mark? C. Two marks and one mark are three marks. T. You may take two pencils in one hand and one pencil in the other hand. T. How many pencils have you in both hands? C. I have three pencils in both hands. T. Two pencils and one pencil are how many pencils ? C. Two pencils and one pencil are three pencils. T. (Writing upon the blackboard, II and I are III.) You may read what I have written. C. Two and one are three. T. (Writing upon the blackboard, 2 and 1 are III.) You may read what I have written. C. Two and one are three. T. I shall now show you another way to make three. 3. T. (Writing upon the blackboark, 2 and 1 are 3.) You may read what I have written. C. Two and one are three. T. (Holding up one block.) How many have I ? C. You have one. T. (Picking up two with the other hand.) How many have I in this hand? C. You have two in that hand. T. How many have I in both hands? C. You have three in both hands. T. One and two are how many ? C. One and two are three. T. You may bring me one splint. ONE TO TEN. 17 T. You may now bring me two splints. T. How many splints did you bring me both times ? C. I brought you three splints. T. (Writing 1 and 2 are) What other word shall I write? C. Three. T. (Completing the sentence with 3.) You may read it now. C. One and two are three. T. (Placing one block on the table.) How many blocks are on the table? C. There is one block on the table. T. (Picking up one with the right hand.) How many have I in my hand? C. You have one in your hand. T. (Picking up another with the other hand.) How many have I in this hand? C. You have one in that hand. T. How many are there on the table and in my two hands ? C. Three. T. One and one and one are how many ? C. One and one and one are three. T. You may bring me one pencil. T. (Holding the object brought.) You may now bring me one splint. T. (Holding this object in the other hand.) You may now bring me one tooth-pick. T. (Having all three dissimiliar objects in full view of the class.) How many things did you bring me? C. I brought you three things. T. One and one and one are how many ? C. One and one and one are three. T. (Writing 1 and 1 and 1 are) What other word shall I write ? C. Three. T. (Completing the sentence with 3.) You may now read what I have written. C. One and one and one are three. T. Two and one are how many? C. Two and one are three. T. One and two are how many ? C. One and two are three. 18 THE SUPPLY. T. One and one and one are how many ? C. One and one and one are three. T. (Holding up three blocks.) How many blocks have I? C. You have three blocks. T. (Taking one away.) How many have I taken away? C. You have taken away one. T. How many are left? C. Two. T. One taken away from three leaves how many? C. One taken away from three leaves two. T. You may bring me three marbles. T. You may give this one to Hattie. T. How many did you bring me? C. I brought you three. ] T. How many did you give to Hattie ? C. I gave one to Hattie. T. How many have I left? C. You have two left. T. Three less one are how many ? C. Three less one are two. T. You may make three marks on the blackboard. T. You may now erase one. T. How many are there now ? C. There are two. T. How many did you erase? C. One. T. Then three less one are how many ? C. Three less one are two. T. (Writing 3 less 1 are 2.) You may read what I have written. C. Three less one are two. T. (Holding up three blocks.) How many have I? C. You have three. T. (Taking two away.) How many have I taken away? C. You have taken away two. T. How many have I left? C. You have one left. T. You may place three blocks on your desk. T. You may now put two of them in your desk. T. How many have you left? ONE TO TEN. 19 C. I have one left. T. Then three blocks less two blocks are how many blocks ? C. Three blocks less two blocks are one block. T. You may make three marks on your slates. T. You may erase two of them. T. How many are left? C. One. T. Three less two are how many ? C. Three less two are one. T. (Writing 3 less 2 are) You may tell me what word to write last. C. One. T. (Writing 3 less 2 are 1.) You may read what I have written. C. Three less two are one. T. You may make three marks on the blackboard. T. You may erase three of them. T. How many are left? C. None. T. You may bring me three pieces of crayon. T. I shall place three of them on Mary's desk. T. How many have I left? C. None. T. Three less three are how many ? C. Three less three are none. T. (Writing 3 less 3 are) What word shall I write last? C. Naught. T. (Writing 3 less 3 are 0. ) You may read. C. Three less three are naught. T. Three less one are how many ? C. Three less one are two. T. Three less two are how many? C. Three less two are one. T. Three less three are how many ? C. Three less three are naught. T. Place one pencil on the table. T. Place another one with it. T. Place another one with them. T. How many ones have you placed on the table ? C. Three ones. 20 THE SUPPLY. T. How many pencils have you placed there? C. Three pencils. T. Three ones are how many? C. Three ones are three. T. Here are some pieces of crayon; all who can tell me how many pieces I have may raise their right hands. T. How many are there? C. There are three. T. Mary, you may take them and tell us how many ones you find. M. I find three ones. T. How many ones are there in three? C. There are three ones in three. T. You may place three splints in a pile. T. You may now see how many twos you can find. T. How many twos have you, Robert? R. I have one two. T. Has any one more than one two ? C. I have one splint more than one two. T. How many have one two and one more? C. I have. T. Then how many twos are there in three? C. There is one two and one in three. T. Three ones are how many ? C. Three ones are three. T. How many ones in three? C. There are three ones in three. T. How many twos in three? C. There is one two, and one more, in three. T. James brought me two roses and John brought me one rose; hdw many roses did both bring me? C. Both brought you three roses. T. Mary has three apples and gives one apple to James; how many apples has Mary left? C. Mary has two apples left. T. John has three cents and spends two cents for candy; how many cents has he left? C. John has one cent left. T. How many sticks of candy at one cent a stick can you buy for three cents? ONE TO TEN. 21 C. I can buy three sticks of candy. T. Mary has one book and Helen has two books; how many books have both ? C. Both have three books. T. Delia bought three oranges, giving one cent for each orange; how many cents did they cost her? C. They cost her three cents. T. Samuel's mother gave him three cents with which to buy bananas at two cents a piece; how many bananas did he buy? C. He bought one banana and had one cent left. T. Jessie bought three peaches and gave two of them to her sister; how many had she left? C. She had one left. T. James has a cat, a dog, and a rabbit; how many pets has he? C. He has three pets. T. Two and how many are three ? C. Two and one are three. T. One and how many are three? C. One and two are three. T. One and one and how many are three ? C. One and one and one are three. T. Three less how many are two ? C. Three less one are two. T. Three less how many are one? C. Three less two are one. T. Three less how many are naught? C. Three less three are naught. T. Three ones are how many ? C. Three ones are three. T. How many ones in three? C. There are three ones in three. T. One two and how many more in three? C. There is one two and one more in three. T. Read: 2 and 1 are 3. C. Two and one are three. T. This is also written, 2 + 1 = 3. T. Then what is this: +? C. It is "and." T. What is this: =? > THE SUPPLY. C. It is "are." T. You may now read this: 1 + 2=3. C. One and two are three. T. You may read : 1 + 1 + 1 =-- 3. C. One and one and one are three. T. How many pints in a quart ? C. There are two pints in a quart. T. What part of a quart is one pint ? C. One pint is one-half of a quart. T. How many quarts in three pints ? C. In three pints are one quart and one pint. T. What do people buy by the quart ? C. They buy milk by the quart. T. (Holding up a foot-rule.) Who can tell me how long this is? Albert. It is one foot long. T. That is right; this is one foot. T. (Holding up a yard-stick.) Who can tell -what I have now? T. It is a yard. T. What is it? C. It is a yard. T. (Holding up the foot-rule again.) What is this? C. It is a foot. T. Now count how many feet there are in a yard. T. (Applying the foot to the yard three times. ) How many did you count? C. I counted three. T. Then how many feet in a yard ? C. There are three feet in a yard. T. Who can tell me something that people buy by the yard ? Florence. They buy cloth by the yard. T. Yes; people buy cloth by the yard. T. What is this? C. It is a foot. T. What is this? C. It is a yard. T. How many feet in a yard? C. There are three feet in a yard. T. What do we buy by the yard? ONE TO TEN. 23 C. We buy cloth by the yard. T. You may show me the pint, William. T. You may show me the quart, Ida. T. How many pints in a quart ? C. There are two pints in a quart. T. One pint and two pints are how many quarts? C. One quart and one pint. T. At one cent a pint what will one quart of milk cost ? C. One quart of milk will cost two cents. T. At one cent a foot what will one yard of ribbon cost ? C. It will cost three cents. T. John has three cents and spends two cents for candy; how many cents has John left? C. John has one cent left. T. James brought me two roses and John brought me one rose; how many roses were brought me? C. Three roses were brought you. T. Here, John, is a three cent piece; you may buy candy with it at one cent a stick; how many sticks of candy can you buy with the three cents? John. I can buy three sticks of candy with the three cents. T. Minnie, I shall give you three cents and you may buy oranges at two cents a piece; how many oranges can you buy? Minnie. I can buy one orange and have one cent left. The preceding models are fairly representative of the character and amount of work necessary in teaching the respective com- binations therein developed. The same general plan is followed in teaching each of the numbers from one to ten inclusive. Of course, the live teacher will understand that the plan as herein outlined connot be strictly adhered to. One class will need more and another class may require less drill to produce a thorough conception of the subject. The questions too must be addressed to the members of the class individually, or to the class collectively, in accordance , with circumstances; no set formula can be mechanically followed. The teacher must know human nature in general, and that of his class in particular; then, with wisdom and tact such as the successful teacher must needs possess, he will adapt, rather than rigidly adopt, methods. The purpose of adaptation is the holding of interest. Retain the 24 THE SUPPLY. interest; regain the interest; or change the subject. It is better to do the last before the second becomes necessary. A close study of the plan outlined discovers that it is purely inductive, that it is developmental, and that it is exhaustive. It will also be seen that the plan incidentally, yet necessarily, comprehends in its scope a most thorough and continuous course in language. All answers are sought to be made in complete sentences that are precisely responsive to the questions asked. This is thought to be essentially helpful to the clear conception of the number idea, and therefore aids both mathe- matically and linguistically. At the conclusion of the develop- ment of the numbers from one to ten inclusive, all the combina- tions within those limits should be made by the pupils, as it were, automatically; that is, results should be announced without the slightest hesitation. A year, of ten months, is conceded to be none too long to fulfill these demands. Much sooner, some- times, it seems that the pupil should be given additional lessons, but as the subsequent work will be easy or difficult in accordance with the facility with which the pupils announce results in this elemental period, it certainly will be advantageous to ' ' make haste slowly." The applied work can be varied without limit for an indefinite period of time, and consequently there is no danger of flagging interest. The work, of course, is almost entirely oral during the first year; yet much drill of the following nature should daily be upon the blackboard, and should be read and completed re- peatedly by the pupils to secure familiarity with such signs and symbols as are in common use: 1+1= 11 = 1x1= 11 = 4- of 2 = 1+1+1= 2- _3 _2 J; 2x5+1= 40 THE SUPPL Y. 2+3+5+1= 4+2+2+3= 7+46= 11 11 11 11 11 11 11 11 8 + 3-2=: -4 3 -^6 -7 5 9 -10 -1. 6+57 = 8+3-5 = In like manner place two splints with the bundle, and ask the pupil to write on the board what you have shown him. He will doubtless write 12 quickly, since he knows where to write the 10 and where the 2; it then only remains to give the number a name. Then develop the number fully, in accord- ance with the plan followed with the eleven. This plan should be continued in the same general manner with the other num- bers to twenty; after which the numbers should be counted from one to twenty, forward and backward, concretely and abstractly, by ones, by twos, by threes, by fours, and by fives, beginning for the twos at one and two indifferently; for the threes at one, two, and three, indifferently; for the fours at one, two, three, and four, indifferently; and for the fives at one, two, three, four, and five, indifferently. Such combinations of two numbers less than ten as will produce a number be- tween ten and twenty, (as for instance, six and five, eight and four, seven and five, nine and four, seven and six, nine and eight) require to be made so familiar that the answer becomes merely a mechanical vocalization of the concept drawn from the visible or audible percept. The mode of reaching this result may be by tabulations, as follows, which should be made by the pupils: 11=1 + 10. 12=2 + 10. 13=3 + 10. 14=4+10. 11=2+9. 12=3+9. 13=4+9. 14=5+9. 11=3 + 8. 12=4+8. 13=5+8. 14=6+8. 11=4+7. 12=5 + 7. 13=6+7. 14 = 7 + 7. 11=5+6. 12=6+6. 15=5+10. 16=6+10. 17 = 7 + 10. 18=8+10. 19 = 9+10. 15=6+9. 16 = 7+9. 17=8+9. 18=9 + 9. 15 = 7 + 8. 16 = 8+8. From the foregoing, the pupils may make subtraction tables as follows : 111 = 10. 122=10. 133=10. 144=10. TEN TO TWENTY. 41 11-2=0. 12-3=9. 13-4=9. 145=9. 11-3=8. 124=8. 135=8. 146=8. 114=7. 12-5 = 7. 136=7. 14-7 = 7. 115=6. 126=6. 137=6. 148=6. 116=5. 127=5. 13-8=5. 149=5. 11-7=4. 12-8=4. 139=4. 14-10=4. 118=3. 129=3. 1310=3. 119=2. 1210=2. 11-10=1. 155=10. 166 = 10. 177 = 10. 188 = 10. 199 = 10. 15-6=9. 16-7=9. 178=9. 18-9=9. 1910=9. 157=8. 168=8. 179=8., 1810=8. 15-8=7. 169 = 7. 1710=7. 15-9=6. 16-10=6. 1510=5. Then such combinations as the following may be tabulated by the pupils: 12=2x6. 14=2x7. 15=3x5. 16=2x8. 18=2x9. 12=3x4. 14 = 7x2. 15=5x3. 16=4x4 18=3x6. 12=4x3. 16=8x2. 18=6x3. 12=6x2. 18=9x2. 20=2x10. i 20=4x5. 20=5x4. 20 = 10x2. 12-5-2=6. 14-2 = 7. 15^-3=5. 16-2=8. 18-5-2=9. 12^-3=4. 14-i-7=2. 15-f-5=3. 16-!-4=4. 18-^3=6. 12-s-4=3. 16-i-8=2. 18-s-6=3. 12-5-6 = 2. 18-^9=2. 20n-2 = 10. 20-i-4=5. 20-f-5=4. 20-s-10=2. 11=2x5+1. 13=2x6+1. 17=2x8+1. 19=2x9+1. 11=3x3+2. 13=3x4 + 1. 17=3x5+2. 19=3x6+1. 11=4x2+3. 13=4x3 + 1. 17=4x4 + 1. 19=4x4+3. 42 THE SUPPLY. 11=5x2 + 1. 13-5x2+3. 13=6x2 + 1. 17=5x3 + 2. 17=6X2 + 5. 17 = 7x2 + 3. 17=8x2 + 1. 19=5x3+4. 19=6x3 + 1. 19 = 7x2 + 5. 19=8x2 + 3. 19=9x2 + 1. The following indicates the character of the work that should also receive much attention at this time: 157581 10 10 30 20 333294 20 10 40 10 420616 30 20 10 30 7 9 8 3 2 3 40 30 10 20 9 9 10 11 12 14 5 3 -4 -7 5 6 Applications like the following should each day be given the pupils, illustrating with the measures themselves, so that as the work goes on the tables of compound numbers shall be uncon- sciously learned: There are two pints in one quart; how many pints are there in two quarts? There are seven days in one week; how many days are there in two weeks? How many cents in two five-cent pieces? How many cents in three three-cent pieces? How many cents in one dime? In one month there are four weeks; how many weeks in two month ? With five cents I bought two oranges at two cents a piece: how many cents did I have left? In one gallon are four quarts; how many quarts in two gallons? In one dime are ten cents; how many cents in three dimes? TWENTY TO ONE HUNDRED. 43 2O TO 1OO. "As a rule the poorest teachers talk most." Sheib. In order that the concepts of the numbers between twenty and thirty, thirty and forty, forty and fifty, etc., shall be per- fect, the bundles of ten each should be associated with the single splints until a few numbers, as twenty-one, twenty-two, twenty-three, and twenty-four, are taught, when the same method may be employed for thirty-one, forty-one, and fifty- one, after which the objects may be wholly dropped, except in the case of' a dull or confused pupil for whom it may be necessary to return to the objects occasionally. The numbers from twenty to one hundred may be, and in fact should be, taught simultaneously. After a clear idea of the units and tens that make up the numbers is acquired, counting by tens from ten to one hundred should be reviewed, so that while counting otherwise no omissions or reduplications shall occur. Counting by ones to one hundred forward and backward, then by fives in Mike manner, then by elevens in like manner, then by twos in like manner, should be added. These are all very simple combinations that will not tax the minds of the pupils severely at all. This work should be followed by tabulations as follows : 1 + 2=3. 2 + 2-4. 3 + 2-5. 4 + 2=6. 11 + 2 = 13. 12 + 2 = 14. 13+2 = 15. 14 + 2 = 16. 21 + 2 = 23. 22 + 2 = 24. 23 + 2 = 25. 24 + 2=26. 31 + 2=33. 32 + 2=34. 33 + 2 = 35. 34 + 2=36. Etc. to Etc. to Etc. to Etc. to 91 + 2=93. 92 + 2=94. 93+2=95. 94 + 2=96. 44 THE SUPPLY. 5+2 = 7. + 2 = 8. 7 + 2=9. 8 + 2 = 10. 9+2=11. Etc. to Etc. to Etc. to Etc. to Etc. to 95+2=97. 96 + 2=98. 97+2=99. 98+2=100. 89+2=91. 1+3=4. 2+3=5. 3+3=6. 4+3=7. Etc. to Etc to Etc. to Etc. to 91+3=94. 92+3=95. 93+3=96. 94+3=97. 5+3=8. 6+3=9. 7 + 3 = 10. 8+3 = 11. 9+3 = 12. Etc to. Etc. to Etc. to Etc. to Etc. to 95+3=98. 96+3 = 99. 97+3 = 100. 88+3=91. 89+3=92. Etc. to 1 + 9 = 10. 2+9 = 11. 3 + 9=42. 4+9=13 Etc. to Etc. to Etc. to Etc. to 91+9=100. 82+9=91. 83+9=92. 84+9=93. 5 + 9=14. 6+9 = 15. 7+9 = 16. 8 + 9 = 17. 9 + 9 = 18. Etc. to Etc. to Etc. to Etc. to Etc. to 85+9=94. 86 + 9=95. 87+9=96. 88 + 9=97. 89+9=98. These may be followed by tabulations as follows 11=0. 21=1. 31=2. 10-1=9. 111=10. 121 = 11. 131=12. 201=19. 211 = 20. 221=21. 231=22. Etc. to 301=29. Etc. to Etc. to Etc. to Etc. to 911=90. 921=91. 931=92. 1001=99. 22=0. 32=1. 102=8. 112=9. 122=10. 132=11. 202 = 18. 212=19. 222=20. 232=21. Etc. to 302=28. Etc. to Etc. to Etc. to Etc. to 912=89. 922=90. 932=91. 1002=98. < 33=0. 10-3 = 7. 11-3=8. 123=9. 133=10. 20-3 = 17. 213=18. 223 = 19. 233=20. Etc. to 30-3=27. Etc. to Etc. to Etc. to Etc. to 913=88. 923=89. 933=90. 1003=97. 104=6. 11-4 = 7. 124=8. 134=9. 204=16. TWENTY TO GLYA' HUNDRED. 45 214 = 17. 224=18. 23-4=19. Etc. to 30 4=20. Etc. to Etc. to Etc. to Etc. to 914=87. 924=88. 934=89. 100 4=90. 10-5=5. 11-5=0. 125=7. 13-5=8. 205 = 15. 21 5 = 10. 225 = 17. 235 = 18. Etc. to 30-5=25. Etc. to Etc. to Etc. to Etc. to 915=80. 925 = 87. 935=88. 1005=95. 100=4. 110=5. 12-0=0. 13-0 = 7. 20-0=14. 21-0 = 15. 220 = 10. 230 = 17. Etc. to 300=24. Etc. to Etc. to Etc. to Etc. to 910=85. 920=80. 930=87. 1000=94. 10-7 = 3. 11-7-4. 127=5. 13 7-= 0. 20 7 13. 217 = 14. 227-15. 237-10. Etc. to 307=23. Etc. to Etc. to Etc. to Etc. to 917-84. 927 85. 937 -80. 1007=93. 10-8-2. 118=3. 128 4. 13-8-5. 20-8 = 12. 21-8 -13. 228=14 23-8 = 15. Etc. to 308 = 22. Etc. to Etc. to Etc. to Etc. to 918 83. 928=84. 938 = 85. 1008 = 92. 109=1. 11-9-2. 129=3. 139-4. 20-9 = 11. 219-12. 229-13. 23-9-14. Etc. to 30 9 =21. Etc to Etc. to Etc. to Etc. to 919 = 82. 929 = 83. 939^84. 100 9 -- = 91. 1010 0. 11-10 1. 12-10-2. 13-10-3. 20-10-10. 21 10 - 1 1 . 22 10 - 12. 2310 1 3. Etc. to 30 10 20. Etc. to Etc. to Etc. to Etc. to 9110 = 81. 9210 82. 9310=83. 100 10 90. 46 THE SUPPL Y. A close study of all these tabulations should be made with the object all the time in view of establishing firmly in the minds of the pupils that, for instance, four and nine are thirteen, fourteen and nine are twenty-three, twenty-four and nine are thirty-three, or in other words, that four and nine added pro- duce a three in units' place, whether they be simple units or units and tens united. To this end, these tabulations should be on the blackboard, or open chart, in full view of the class at all times; and much special attention should be devoted to these com- binations each day. Certain combinations must be given very much more attention than others; for instance, nine and seven is always troublesome and should receive much attention; three and two is very simple and therefore wiH require very little attention. But all must be dwelt upon until results are given mechanically. This above all others is the place for thorough work. So long as any incorrect results are announced by the pupils, so long must no thought of leaving the subject enter the mind of the teacher or the pupil. Ninety-nine mistakes in a hundred in the computations in arithmetic are the result of imperfect knowledge of addition; in these tabulations is all addition; hence the conclusion, that these concepts must be faultless, that _is, drawn from faultless percepts. These thoughts must be applied to column addition, the real work of life for which the pupil is being fitted, if the public school system is attaining its true ends. Following the counting by tens, by ones, by twos, by fives, and by elevens, should be formed multiplication tables of the same with one to ten inclusive, (and from them division tables should be formed) by the pupils themselves, as follows: Lxl=l 1x2 = 2 1x10 = 10 1x5=5 1x11 = 11 2x1=2 2x2=4 2x10=20 2x5 = 10 2x11 = 22 3x1=3 3x2=6 3x10=30 3x5 = 15 3x11=33 4x1=4 4x2 = 8 4x10=40 4x5 = 20 4x11=44 5x1=5 5x2 = 10 5x10=50 5x5=25 5x11=55 6x1=6 6x2 = 12 6x10=60 6x5=30 6x11=66 7x1 = 7 7x2 = 14 7x10 = 70 7x5=35 7x11 = 77 8x1=8 8x2 = 16 8x10=80 8x5=40 8x11=88 9x1=9 9x2=18 9x10 = 90 9x5=45 9x11=99 10x1=10 10x2=20 10x10=100 10x5=50 TWENTY TO ONE HUNDRED. 47 These tables may be read interchangeably, two times ten are twenty, two tens are twenty, two tens in twenty. The last form will be all the change that it will be necessary to make in passing from multiplication to division. The following will indicate the co-ordinate work at this time: Addition: 8 12 27 21 18 16 52 <>5 6 2 7 13 33 19 25 12 19 8 27 85 (> 21 25 20 17 10 9 8 9 3 4 32 12 40 31 4<> 12 14 40 6 Subtraction : 17 13 16 _9 _7 _8 Multiplication: 4 12 1 1 12 9 8 13 14 5 48 _1 21 '> 19 10 25 12 43 24 39 4(5 38 6 10 12 20 18 16 14 15 5 5 5 5 5 5 5 5 2 3 468 230 10 10 10 10 10 11 11 11 5 48 791 11 11 11 11 11 11 Division: 1)5 2)4 5)15 10)40 11)33 2)16 5)35 1)9 Applications: Jane went to the store and bought an orange for three cents, some candy for five cents, and some peanuts for four cents; what did all cost? In one gallon are four quarts; how many quarts in five gallons? John's father gave him ten cents with which to buy a whistle for five cents, and to bring back the change; how much did he bring back ? 48 THE SUPPLY. I gave Henry a half-dollar with which to buy tive pounds of sugar at five cents a pound; how much change should he bring me ? What part of a pound of sugar are eight ounces? Thirty minutes are what part of an hour? Twenty minutes are what part of an hour? Fifteen minutes are what part of an hour? How much is a peck of potatoes worth at forty cents a bushel? A man riding a bicycle travels ten miles per hour; in how many hours can he travel to Sacramento, a distance of one hundred miles? Ten boys wish to fire a pack of fire-crackers so that each may fire the same number; the pack contains one hundred; how many should each one fire? Allie came to school with fifty walnuts; he gave each of his ten playmates four walnuts; how many had he left? Counting by fours from one to one hundred, beginning with four, by threes in like manner, by sixes, by eights, by nines, and by sevens, should now receive daily attention in the order given. This should be done step by step. The fours should be thoroughly mastered before the threes are taken up. The counting should be both forward and backward. When taken in this order, the severity of the work does not vary much at any two consecutive steps. The work will be found to be graduated on the basis of association and growth. It is much more difficult to count by threes than by twos; but with the graduated progress from two to three, here given, the three at its time will present no more difficulties than the two at its time. ONE HUNDRED TO ONE THOUSAND. 1OO TO 1OOO. "Pythagoras taught that number is the first principle of existence. ' ' Hittell. It will not be necessary to devote much attention to the numbers from one hundred to one thousand if those below one hundred have been as thoroughly taught as they should be; for the mind of the pupil is so well drilled in regard to the relative value of units and tens that only a brief time will be required to establish the same relationship between tens and hundreds. A single lesson will often be sufficient to enable all pupils to read any number from one hundred to one thousand. In addition, subtraction, multiplication, and division, there is really nothing new; simply an extension of the identical knowl- edge which the pupil already possesses. The work therefore is simply expansive in its nature and in the rapidity and accuracy of its execution. Much work of the following character, with great attention to accuracy and rapidity, must be given the pupils every day: Addition : 198 10 568 135 532 169 32 358 139 90 109 132 9 25 85 200 26 14 25 126 9 76 8 314 75 85 19 32 7 216 432 39 110 91 165 100 Subtraction: 865 269 132 820 736 893 132 157 129 184 492 98 Multiplication: 231 195 263 84 36 74 222 5 10 11 Division: 2)648 5)250 10)980 11)451 50 THE SUPPLY. Tabulation should be continued as follows: 1x4: 2x4: 3X4: 4X4 = 5X4: 6X4: 7X4: 8X4: 9x4 = 10X4: A. =8. --12. =16. = 20. =24. =28. -32. =36. =40. 1x9 = 2x9: 3x9: 4x9: 5x9: 6x9: 7x9: 8X9: 9x9: 10X9: 1x3-3. 2x3=6. 3x3=9. 4x3=12. 5x3=15. 6x3=18. 7x3=21. 8x3=24. 9x3=27. 10x3=30. :9. =18. = 27. =36. =45. =54. =63. = 72. =81. =90. 1X6: 2x6 = 3x6: 4X6: 5x6 = 6x6: 7X6 = 8x6: 9x6 = 10X6: :24. :30. =36. =42. =48. 54. 60. 1X7: 2X7: 3X7: 4X7: 5x7: 6x7: 7X7: 8X7: 9X7: 10X7: 1x8=8. 2x8=16. 3x8=24. 4x8=32. 5x8=40. 6x8=48. 7x8=56. 8x8=64. 9x8=72- 10x8=80. = < . =14. = 21. = 28. = 35. =42. 49. =56. =63. =70. Retain the readings, two times three are six, two threes are six, two threes in six. The order of the tables, one, two, ten, five, eleven, four, three, six, eight, nine, seven, will be found to be the natural method of presentation, because the pupil will learn to count by ones most readily; by twos, tens, and fives, with little or no difficulty; by elevens nearly as readily; by fours more readily than by threes, as the digits in units' place, four, eight, two, six, naught, constantly repeat; then by threes and by sixes, as being next in order and naturally associated; by eights, as being measurably associated with fours; by nines, as being associated with threes somewhat; and lastly, by sevens, as being most difficult, having no association with the others and no regularity of scale which can be pointed out to the young pupil with advantage. Applications should be continued as follows: There are seven days in one week; how many days in six weeks ? How many cents in a dollar? How many cents in a dime? How many cents in a half-dollar? ONE HUNDRED TO ONE THOUSAND. 51 How many cents in three dimes? How many cents in a nickel? How many cents in two five-cent pieces? There are four quarts in one gallon; how many quarts in six gallons? There are three feet in one yard; how many feet in eight yards ? One boy has eight fingers; how many fingers have five boys? John has two cents and James has one-half as many; how many has James? What is one-half of four? How many cents in one dime? How many cents in one half-dime? How many pints in one quart? How many pints in one-half of a quart? I gave Julia six cents and Mary one-half as many; how many did I give Mary? Hattie brought me three- pinks and Lily brought me two less than Hattie; how many did both bring me? CO-ORDINATE EXERCISES. 4- Of 2= = 2-s-2= 3-5-3 = 5-5 = .Vof4= | of 6= ioflO = 4-5-2= 6-5-3= 10--5= 4- of 6= iof9 = of!5 = 6-i-2= 9-t-3 = 15-5-5 = of8= fofl2= \ of 20= 8-*-2 = 12--3= Etc. to 20-i-5= 4- of 10= of!5 = -I of 23 = 102= 15-s-3= 25+5= Etc. to Etc. to Etc. to 4-of20 = i of 30= } of 50= 20-s-2= 30-i-3= 505 = 128 703 862 83 93 9 -195 xll 10)840 75 83 46 12 708 95 28 109 95 X5 5)625 THE SUPPLY 31 20 105 500 -180 87 X2 11)891 Preparatory to ready and thorough work in " Long Division," in Factoring, and in Greatest Common Divisor and Least Com- mon Multiple, the following chart should be made by the class under the direction of the teacher: 1 o 3 4 5 8 7 8 9 10 11 12 . 13 14 15 16 17 18 9 4 8 8 10 12 14 16 18 20 2-2 2-1 2U 28 30 32 31 36 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 4 8 12 16 20 24 28 32 36 40 44 48 52 .")(i 60 64 68 72 6 10 15 20 25 30 35 40 45 .50 55 6!) 65 70 75 80 85 90 (i 12 18 24 30 36 42 48 54 60 66' 72 78 84 90 96 102 108 i 14 21 28 35 42 49 56 63 70 77 84 91 98 105 112 119 I2(i 8 16 24 32 40 48 56 64 72 80 88 06 104 112 12'1 128 13(i 1-14 9 18 27 36 -15 54 63 72 81 90 99 los 117 126 1.35 144 153 162 10 20 30 40 50 60 70 80 90 100 110 12. > 130 140 150 160 170 180 11 22 33 44 55 66 77 88 99 110 121 132 148 187 12 24 86 48 60 72 84 96 108 12) 132 144 13 26 89 52 65 78 91 104 117 130 143 169 221 14 28 42 58 70 84 9.S 112 12i 1-10 196 15 80 45 BO 75 80 105 120 135 1.50 225 16 32 4* 64 SO M 112 12-! 141 160 256 17 84 51 68 65 102 11!) 136 153 170 1S7 221 289 18 88 54 72 !() 10S 126 144 162 180 824 10 38 57 76 95 114 133 152 171 191) 219 247 323 ONE THOUSAND AND Aim YE. 1OOO AND ABOVE. "It is comparatively rare to find a candidate who can add correctly a long column of figures." Civil Service Examiners. In order that the true conception of the method of reading and writing large numbers may be obtained by the pupils, it will be found simplest to make inclosures of some kind in which the numbers may be written by periods of three figures each, each inclosure, or period, being read as if standing alone, the pupils attaching to the reading of the number in the period the name of the enclosure, or period. The following will illustrate the method and convince the reader of its simplicity; 865 1000 85 5H5 732 840 4 625 835 Much practice should be given to the reading of the numbers in the several boxes, or periods, separately at first, having definite names for the boxes in accordance with their location. The right hand box may be left nameless, as the numbers therein are always read without the box, or period, name. The numbers in the second box (thousands) should be read without any reference to the other boxes: that is, the 85 of the 54 THE A UPPL Y. number S5,5(>5 should be read as eighty-five, and the word "thousands" should attach simply as the word "horses" would attach to the number. In this way the reading of numbers is very simple, as no numeration is ever indulged in to retard rapidity of execution. After this process of reading is carried on until numbers are read without the least hesitation, the box names may be removed and the readings continued until the names are fixed by location, when the boxes themselves may be omitted and the comma used in their stead. The comma should never be 'omitted afterward until both the reading and the writing of all numbers is as rapid and accurate as possible. After the reading of numbers in the several steps given, the pupil should be given much practice in the writing of numbers in the several steps successively. The method indicated will obviate the difficulty and doubt encountered by all in knowing when to insert ciphers and how many, and will certainly save much labor and induce accuracy and rapidity. Work now should broaden out to its fullest in both addition and subtraction. The subject ot addition should at all times be given the most careful attention; pupils should be taught to add upward and downward with equal and great facility, naming results only as they proceed. The time devoted to ad- dition as compared to that devoted to subtraction must be, thoughtfully speaking, as ten to one; and nearly the same ratio will be found to exist between the time required for ad- dition and that required for either multiplication or division. Nearly all errors in school and among business men are directly or indirectly the result of imperfect work in addition. Pupils should add long columns, with a view to accuracy first and rapidity secondly, every day of their school lives until they are twelve years old, and even then the exercise must not be allowed to fall into any considerable degree of disuse. Multiplication by numbers containing two or more digits should also be practiced at this stage of advancement, and clear ideas in regard to the beginning of a partial product one place farther to the left than its immediate predecessor should be inculcated. Short division of all numbers should be given much attention. The following indicates the character of the co-ordinate work at this time: ONE THOUSAND AND ABOVE. 55 Addition: 8,963 89,645 986,746 456,834 84 7,054 38,395 300,846 904,386 36 765 26,283 258,984 784,666 92 392 13,725 98, 642 292,564 78 843 185.939 89,206 964,238 43 6. 483 205.684 1.864.684 792. 189 56 29 46 92 _73 Subtraction : 189,425 40,389 $36,472 4,007,392 74.932 19,426 293. 783 1.006.829 8,965,432 7,064,584 34,856,329 867,345,732 2.893,876 2.092.864 19.857.184 193.845.783 Multiplication: 64.530,243 894,320 5,004,234 74,385 25 360 125 170 986.432,894 846.785 2,345,860 93,656,487 432 79_ 280 564 Division: 6)482. 356 8)8.375.493 9)84.293.008 7)483.754 Constant review of tabulations regularly and irregularly, oral and written: and further extensions of tabulations as follows: 1x12 = 12. 1x13 = 13. 1x14 = 14. 1x15 = 15. 2x12 = 24. 2x13 = 26. 2x14=28. 2x15=30. 3x12=36. 3x13=39. 3x14=42. 3x15=45. 4 x 12 = 48. 4 x 13 = 52. 4 x 14 = 56. 4 x 15=60. 5x12=60. 5x13=65. 5x14 = 70. 5x15=75. 6x12 = 72. 6x13 = 78. 6x14=84. 6x15 = 90. 7x12-84. 7x13 = 91. 7x14 = 98. .7x15 = 105. 8x12=96. 8x13 = 104. 8x14 = 112. 8x15 = 120. THE Sr 9x12 = 1 OS. 9 x 1.3 = 117. 9x14 = 1 26. 10x12 = 120. 10x13 = 130. 10x14 = 140, 1x16 = 16, 2x16=32. 3x16=48. 4x16-64. 5x16=80. 6x16=96. 7x16=112. 8x16 = 128. 9x16 = 144. 10x16=160. 9x15 = 185. 10x15 = 150. 1x17=17. 1x18 = 18. 1x19 = 19. 1x20=20. 2x17=34. 2x18 = 36. 2x19=38. 2x20=40. 3x17=51, 3x18=54. 3x19 = 57. 3x20 = 60. 4x17 = 68. 4x18=72. 4x19 = 76. 4x20=80. 5x17=85. 5x18=90. 5x19=95. 5x20=100. 6x17 = 102. 6x18 = 108. 6x19=114. 6x20=120. 7x17 = 119. 7x18=126. 7x19 = 133. 7x20=140. 8x17=136. 8x18=144. 8x19 = 152. 8x20=160. 9x17 = 153. 9x18 = 162. 9x19=171. 9x20 = 180. 10x17 = 170. 10x18=180. 10x19 = 190. 10x20 = 200. The preceding tables should not be directly committed to memory but should be given to the pupils with the last columns blank, and the pupils should be required to fill in the products on their slates or blank books or upon the blackboard. In this way much greater power will be acquired for "long divi- sion" than could be acquired without these exercises, and ac- curacy and rapidity, the watch-words of arithmetic, will be given a wonderful impulse. The applications should now be more comprehensive, and simple explanations should be developed. These applications should include more and more the tables of compound num- bers that are of frequent use in the every day affairs of life. In developing explanations, begin with simple applications like the following: What will' four apples cost at one cent each? Explanation: Since one apple costs one cent, four apples will cost four times one cent, or four cents. ONE THOUSAND AND ABOVE. f>7 Keep the explanations short yet complete. Another: How many oranges can I buy with ten cents, at two cents a piece? Explanation: Since one orange costs two cents, for ten cents I can buy five oranges. This explanation is short and pointed, but is objected to some- times as failing to indicate the process. Later on the clause indicating the process may be introduced. The explanation will then be as follows: Since one orange costs two cents, I can buy as many times one orange for ten cents as two cents are contained times in ten cents, or five times one orange, or five oranges. The following explanation is also a good one, but is open to the objection that it is not consonant with the general process employed by the business world: Since two cents will buy one orange, one cent will buy one-half an orange and ten cents will buy ten times one-half an orange, or five oranges. This explanation is also open to the objection that it involves the use of fractions which are as yet but imperfectly developed. A concise and precise explanation is of double utility; it assists the understanding and begets confidence, and at the same time is incidentally a valuable language-builder. THE SUPPLY. THE PRIME NUMBER. The concept of a prime number is a necessary antecedent to a clear conception of the subjects of Greatest Common Divisor and Least Common Multiple. The method by which this concept is implanted in the mind of the pupil is both simple and un- failing: T. (To class. ) By what may one be divided ? C. By one. T. By what may two be divided? C. By two and by one. 1 4 T. By what may three be divided? 2 <> C. By three and by one. H H T. By what may four be divided? 5 ?> C. By four, by two, and by one. 7 10 T. Do you notice any difference between the four and the three ? C. Yes; four may be divided by some other number than itself and one. T. Then we shall place it in another column. T. By what numbers may five be divided? 11 \"2 C. By five and by one, 13 14 T. Then in which column shall I write it? 17 15 C. In the first column. 1<> T. By what may six be divided? C. By six, by three, by two, and by one. T. Does six belong in the first column then? C. No. T. That is right; and as we will write all numbers in the THE PRIME NUMBER. oi> second column that do not belong in the first column, I shall write this one in the second column. T. By what may seven be divided? C. By seven and by one. T. In which column shall I write it? C. In the first. T. By what may eight be divided? C. By eight, by four, by two, and by one. T. James, you may tell me in which column to write it. J. In the first. T. Lucy, did James answer correctly? L. No; it should be written in the second column. T. Can you tell me why? L. Because it may be divided by other numbers than eight and one. T. Very good. T. Mary, by what may nine be divided? M. By nine, by three, and by one. T. To which column does it belong? M. To the second. T. To which column then do you think ten belongs? M. I think ten belongs to the second column, because it may be divided by ten, by five, by two, and by one. T. Now who can tell to which column eleven belongs? Robert. (Holding up his hand.) It belongs to the first, because it may be divided by eleven and one only. T. That is right. T. The numbers in the first column are called PRIME NUMBERS. T. Now think a moment, and then I want all who think they can tell me what a prime number is to raise their right hands. T. (After waiting a short time.) How many think they can tell me what a prime number is? (Many hands are raised and after many answers more or less perfect.) Robert. A prime number is a number that may be divided, without a remainder, by itself and one only. T. Jane, you may tell me whether twelve is a prime number or not? <>0 THE SUPPLY. (. Twelve is not a prime number, because it may be divided by some other number than itself and one, by six for example. T. Helen, you may tell me whether thirteen is a prime number or not? H. Thirteen is a prime number, because it may be divided by itself and one only. T. Thomas, you may write fourteen, fifteen, sixteen, and seventeen in these columns, being careful to place each num- ber where it belongs. In the same manner all numbers from one to one hundred should be segregated into the two classes, prime numbers and those that are not prime numbers. The prime numbers then should be committed to memory as soon as is convenient, so that errors in greatest common divisor and least common multiple may be avoided. O. C. D. AND L. C. M. in GREATEST COMMON DIVISOR AND LEAST COMMON MULTIPLE. ' ' I assure you there is no such whetstone to sharpen a good wit and encourage a will to learning as is praise." Ascham. The object of developing the idea of a prime number was to prepare the student properly to comprehend the subjects of greatest common -divisor and least common multiple These subjects should be approached only by means of factoring; that is, by considering numbers with reference to the prime numbers that are contained in those numbers an exact num- ber of times. For example, 4-2x2. T. What is a divisor of four? C. Two is a divisor of four. T. What is the greatest divisor of four? C. Two twos, or four, is the greatest divisor of four. Then take two numbers and resolving them into their prime factors: <>-2x.S. 8 = 2x2x2. T. What is a divisor of six? C. Two is a divisor of six. T. What is another divisor of six? C. Three is another divisor of six. T. Is there any other divisor of six? C. Yes; two times three, or six, is another divisor. <;:> THE SUPPLY. T. What is a divisor of eight? C. Two, two times two, or two times two times two. T. Which of these is also a divisor of six? C. Two is also a divisor of six. T. Yes; two is also a divisor of six. Two then is a common divisor of six and eight. Then take three or more numbers that contain more than one common divisor: 6-2 x 8. 12=2x2x3. 18-2x3x3. T. What are the divisors of six ? C. Two, three, and two times -three are the divisors of six. T. What are the divisors of twelve? C. Two, three, two times two, two times three, and two times two times three are the divisors of twelve. T. What are the divisors of eighteen? C. Two, three, two times three, three times three, and two times three times three are the divisors of eighteen. T. Now look carefully and see what will divide each one of the three numbers. C. Two will divide each one of the numbers. T. What else will divide each one of the numbers? C. Three will also divide each one of the numbers. T. Is there any other number beside two and three that will divide each one of the numbers, six, twelve, and eighteen? C. Yes; two times three will also divide them. T. Then, what are the common divisors of six, twelve, eighteen ? C. Two, three, and two times three, or six, are the common divisors of six, twelve, and eighteen. T. Then, what is the greatest common divisor of six, twelve, and eighteen? C. Two times three, or six, is the greatest common divisor of six, twelve, and eighteen. T. Now, looking at the prime factors of six, twelve and eighteen, can you tell me what the greatest common divisor of the numbers contains? C. The greatest common divisor of six, twelve, and eighteen, G. C. D. AND L. C. M. 63 contains all the the prime factors that are found in each of them. T. I wish to see now whether you can find the greatest common divisor of fifteen, twenty-five, thirty-five, and forty-five. C. The prime factors of fifteen are three and five; the prime factors of twenty-five are five and five; the prime factors of thirty-five are five and seven; and the prime factors ~or forty- five are three, five, and five. Five is the only factor that is contained in each of the numbers, fifteen, twenty-five, thirty- five, and forty-five; hence five is the greatest common divisor of fifteen, twenty-five, thirty-five, and forty-five. Blackboard form : 15 = 3 x 5. 25 =5x5. 35^5x7. 45-3x3x5. Therefore 5 = G. C. D. The visible solution should be as brief as possible, consistent \vith a clear understanding of the process. The solution should be accompanied or supplemented by an oral explanation that shall convince both the teacher and the pupils that the pupil clearly understands his work. The idea of the greatest com- mon divisor being developed and clearly and firmly fixed in the minds of the pupils, the subject of the least common mul- tiple should be approached by an analogous process: T. What number contains two an exact number of times? C. Four contains two twice. T. Is there any other number that contains two an exact number of times? C. Yes; six and eight, as well as a great many others. T. What number contains three an exact number of times ? C. Six, nine, twelve, and many other numbers. T. As six contains three an exact number of times, it is called a multiple of three. T. Now what is a multiple of four? C. Eight is a multiple of four; twelve is a multiple of four: many other numbers are multiples of four. T. What number is a multiple of two and also a multiple of three ? C. Six is a multiple of two and also a multiple of three. (>4 THE SUPPLY. T. What did we call a number that is a. divisor of two or more numbers? C. We called a number that is a divisor of two or more numbers, a common divisor. T. Then what shall we call a number that is a multiple of two or more numbers? C. A number that is a multiple of two or more numbers is a common multiple of those numbers. T. Then six is a common multiple of two and three. T. Now, can you think of any other common multiple of two and three ? C. Twelve is another common multiple of two and three; eighteen is another common multiple of two and three. T. What is the least common multiple of two and three? C. Six is the least common multiple of two and three. Take numbers and factor them into their prime factors: 6=2x3. 8=2x2x2. 9=3x3. T. What factors does six contain? C. Six contains two and three. T. Then a number that contains six must contain what? C. A number that contains six must contain the factors two and three. T. A number that contains, eight must contain what? C. A number that contains eight must contain two as a factor three times. T. Then a number that contains both six and eight must contain what ? C. A number that contains both six and eight must con- tain, as factors, three twos and a three. T. What number contains three twos and a three as factors? C. Twenty-four contains three twos and a three as factors. T. Then what is a common multiple of six and eight? C. Twenty-four is a common multiple of six and eight. T. Is there any other number less than twenty-four that contains both six and eight? C. No. T. Then what is the least common multiple of six and eight ? G. C. D. AND L. C. V. tio C. Twenty-four is the least common multiple of six and eight. T. Now look carefully at the three numbers, six, eight, and nine, and tell me what factors the least common multiple of these three numbers must contain? C. The least common multiple of six, eight, and nine must contain three twos and two threes as factors; since to contain eight it must contain three twos, and to contain nine it must contain two threes; to contain the six no additional factors are necessary as already all of its factors have been selected. T. Then, the least common multiple of six, eight, and nine is what ? C. The least common multiple of six, eight, and nine is seventy-two. T. Now you may find the least common multiple of ten, fifteen, twenty, and twenty-five. C. 10 = 2x5. 15=3x5. 20=2x2x5. 25 = 5x5. Therefore 2x2x3x5xo=300=L. C. M. The least common multiple must contain as factors two twos, because it must contain all the prime factors of each number and there are two twos in twenty; it must contain a three in order that it may contain fifteen; and it must contain two fives in order that it may contain twenty-five. It need not contain any other factors, for it now contains all the prime factors of each of the numbers; hence the least common mul- tiple of ten, fifteen, twenty, and twenty-five contains as factors two twos, one three, and two fives, and is therefore three hundred. THE SUPPL Y. FRACTIONS, "Questions must be given to suit all capacities."- Currie. While, in considering whole numbers, we may gain for the pupil some ' idea of fractions, the true idea of fractions is never obtained until the unit is separated into its parts. Usually children before coming to school have an indefinite, and perhaps in a few exceptional cases a definite, idea of certain fractional parts, as halves and quarters; it is very rare that any other fractional parts are familiar to any of them. It will generally be found, moreover, that their idea of a half is only approximate, that is, that they consider that an apple divided into two parts in the ordinary way is divided into halves; hence the use of the apple or of any other irregularly formed object must be supplemented by something more exact in its dimensions. The line upon the blackboard, as it may be measured by the pupil himself, is probably the best object with which to correct the pupil's crude definition of a half, (to-wit: A half is one of the two parts of any thing.) and to induce him to insert the word "equal" between "two" and "parts" and to make other corrections if necessary, giving ultimately the true definition, A half is one of the two equal parts of any thing. The following outline of a first lesson in fractional parts will indicate the character of the work deemed proper, the quantity being left to the good judgment of the faithful an inventive teacher: T. (Holding up an apple and cutting it into two, as nearly as possible, equal parts.) Into how many parts have I cut the apple? FRACTIONS. 67 C. You have cut it into two parts. T. What is each part? C. Each part is one-half. T. . That is right; th'en what is one-half? C. It is one of the two parts of any thing. T. Now look closely and I will take another apple and cut it. (Cutting the apple into two parts quite unequal in size.) Is each one of these parts a half? C. (Answers at first are divided.) T. I will take another apple and cut it. (Cutting it into two parts very unequal in size.) Is each one of these parts a half? C. (Seeing now the great inequality.) No. T. Why not? C. The parts are not equal. T. But you said, One-half is one of the two parts of any thing. Do you wish to correct your definition? C. Yes. T. Very well, you may do so if you will be very careful. Ready. C. A half is one of the two equal parts of any thing. T. That will do very well. T. How many halves in any thing? C. There are two halves in any thing. T. I will write one-half on the blackboard. Q-.) The two under the line is the number of parts into which I divided the apple, and the one is the one part. T. How many halves in one? C. There are two halves in one. T. How many are two times one-half? C. Two times one-half are one. T. How much are one-half and one-half? C. One-half and one-half are one. T. If I take away one-half from one, how much is left? C. One-half is left. T. How shall I write two-halves? C. f. T. If I take two-halves of one apple and one-half of another, how many halves will I take? C. You will take three-halves. OS THE SUPPLY. T. How shall I write three-halves on the blackboard? p 3 L-. TjT. T. How much are two-halves and one-half? C. Two-halves and one-half are tfiree-halves. T. How much are three times one-half? C. Three times one-half are three-halves. T. How many times is one-half contained in three-halves? C. One-half is contained in three-halves three times. T. How much is two-halves? C. Two-halves is one. T. How much is one plus one-half? C. One plus one-half equals one-and-one-half. T. This is the way one-and-one-half is written. (14-. ) T. How much then are two-halves and one-half? C. Two-halves and one-half are one-and-one-half. T. How much then is three-halves? C. Three-halves equals one-and-one-half. The subject of halves may be continued in this manner until the teacher feels satisfied that the subject is understood. The subject of thirds should be presented upon the black- board with the line, in substantially the same manner that the subject of halves was presented with the apple. T. Into how many parts is this line divided? C. It is divided into three parts. T. Measure the parts and tell me into what kind of parts it is divided. C. It is divided into three eq^^al parts. T. What shall we call one of the parts? (The teacher may be obliged to give the name himself.) T. or C. One-third. T. How shall I write it? C. i. T. What is one-third? C. One-third is one of the three equal parts of any thing. The subject of thirds may be continued as the subject of halves was continued. No definite comparisons between the halves and the thirds should be made until the subject of sixths has been considered. FRACTIONS* 68 The subject of fourths should be approached in the same manner as the subject of thirds, and requires the same character of questions and work, with the following in addition: T. Into how many parts have I divided this line? C. Into two parts, T. What is each part? C. Each part is one- half. T. (Using the same line.) Into how many parts have I divided it now? C. Into four parts. T. What is each part? C. Each part is one-fourth. T. How many one-fourths in one-half? C. There are two one-fourths in one-half. T. What is one-half of one-half? C. One-half of one-half is one-fourth. T. One-half equals what? C. One-half equals two-fourths. T. How much are one-half and one-fourth? C. One-half and one-fourth are three-fourths. At this point practical applications in .fractional forms should be introduced that involve the ideas of halves and fourths either separately or conjointly. The following will indicate something of the nature of the applications that may be given advantageously: (Short oral explanations should accompany all applications.) What is the sum of one-half of four cents and one-fourth of four cents ? What is three-fourths of four cents? If a pound of coffee is worth sixteen cents, what is one-fourth of a pound worth ? If a gallon of milk cost twenty cents, what will a quart cost? How many hours in a day? How many hours in one-half of a day? How many hours in one-fourth of a day? 70 THE SUPPLY. How many hours in three-fourths of a day? How many cents in one dollar? How many cents in one- half of a dollar? How many cents in one-fourth of a dollar? How many cents in three-fourths of a dollar? How many ounces in one pound of sugar? How many ounces in one-half of a pound of sugar? How many ounces in one-fourth of a pound of sugar? How many ounces in three-fourths of a pound of sugar ? There are thirty-one and one-half gallons in a barrel; how many gallons in two barrels? There are five and one-half yards in one rod; how many yards are there in two rods? There are four quarts in a gallon; one quart is what part ot a gallon ? Two quarts are what part of a gallon ? Three quarts are what part of a gallon ? The subject of fifths is entirely analogous to that of thirds in the manner of its presentation. The subject of sixths is quite like that of fourths, except that it requires the establishment of the equivalence in sixths of both thirds and halves, as well as more extensive applications in- volving the three fractional units. The following will give a general idea of the manner of de- veloping this additional conception: T. Into how many parts is this line divided? C. Into two parts. T. What is each part? C. One-half. T. Into how many parts is the line divided by the points below the line? I C. Into three parts. T. What is each part? C. One-third. FRACTIONS. 71 T. Into how many parts is the line divided by the short lines ? C. Into six parts. T. What is each part? C. One-sixth. T. How many one-sixths in one-third? C. There are two one-sixths in one-third. T. How many one-sixths in one-half? C. There are three one-sixths in one-half. T. How many times is one-sixth contained in one-third? C. One-sixth is contained in one-third twice. T. How many times is one-sixth contained in one-half? C. One-sixth is contained in one-half three times. T. How many times is two-sixths contained in three-sixths? C. Two-sixths is contained in three-sixths one and one-half times. T. How many times is one-third contained in three-sixths? C. One-third is contained in three-sixths one and one-half times. T. Then how many times is one-third contained in one-half? C. One-third is contained in one-half one and one-half times. T. How much is one-half times three? C. One-half times three equals three-halves, or one and one- half. T. How much is one-half divided by one-third? C. One-half divided by one-third equals one and one-half. T. What is one-half multiplied by three? C. One-half multiplied by three equals three-halves, or one and one-half. T. Then one-half divided by one-third equals one-half mul- tiplied by three, does it? C. It does. T. What is the difference between the two expressions? C. The latter has the three above instead of below the line, and is multiplication instead of division. T. Let us see whether we may always do as we have done in this example. 72 THK S UP PL Y. T. Ho\v much is two-thirds divided by one-half? C. Two-thirds divided by one-half equals one and one-third. T. What is two-thirds multiplied by two? C. Two-thirds multiplied by two equals one and one-third. T. What do you see this time? C. I see that two-thirds divided by one-half equals two- thirds multiplied by two. T. Well, you may try other examples and see whether you can find any in which this is not true. This comparison of results should be continued as other fractional forms are taught until sufficient experience is gained by the pupil to cause him to be certain that questions of that kind can be solved in either manner, when it may be given him as a rule that, In dividing one fraction by another the divisor may be inverted and the methods of multiplication employed. It should also be drawn out what the object of so doing is; to-wit, convenience. Sevenths are developed as were fifths. Eighths are developed as were sixths, halves and fourths both being again considered, but this time with eighths as units of measure. Ninths are developed in the same manner as fourths were, thirds being considered with them. The following will indicate the character of the co-ordinate work at this time: 21)210( = i + i = \~\^ 2xf= 16)320( |= l+f = f-l= 3x1 = 42)1260( | 1+1= 1-1 = 4x1 = 13)86( f= 1+1- |-1= 5x1= 35)99( f= 1+1= 1-1= 3x1 = 41)128( f= |+1= 1-1= 4x1= 93)325( f= 1+1= 1-1= 6xf = 103)842( f= l+f= 2-1= 9x1 = 82)672( ft= |+1= 2-1= 75)567( 1+1= 2-1= 91)635( 1+1= 11-1= 8x| = 107)932( 1+2= 11-1= 7xf= 4-5-2- x2 FRACTIONS. | of 2 = 2-5-2 of 1 i-f 1.1 ?~ 1x3 All work in fractions should be diagramed by the pupils until they are perfectly clear in their comprehension of the principles of fractions; then the use of diagrams should be discontinued. The following will indicate the character of the diagram that will produce satisfactory results: -s-^=f. Same as above. xf=f. Should not be diagramed because of its simplicity. of 2 -4. The first diagram will also illustrate questions like the follow ing: How many times is one-fourth contained in one- half? How many times is one-half contained in one-fourth? How many times is one-half contained in three-fourths? 74 THE SUPPL Y. How many times is three-fourths contained in one-half? The second diagram will also illustrate questions like the fol- lowing: How many are one-half and one-third? How many are one-half and one-sixth? What is the difference between one-half and one-sixth ? What is the difference between two-thirds and one-half? What is one-half of one-third? What is one-third of one-half? What is two-thirds of one-half? How many times is one-sixth contained in one-half? How many times is one-third contained in one-half? How many times is one-half contained in one-third? How many times is two-thirds contained in one-half? How many times is five-sixths contained in two-thirds? FRA CTIONS--DECIMA LH. FRACTIONS DECIMALS The subject of tenths introduces to the pupils new percepts for the same concepts under given circumstances. The manner of treating tenths is precisely the same as that of treating any other fractional concept, except that when the name is drawn from the pupil and its numerical equivalent written, it is to be written first as ^ and secondly as .1. The manner of proceeding may be as follows: T. (Dividing a line into ten equal parts.) Into how many parts is this line divided? C. Into ten equal parts. T. What is each part? C. Each part is one-tenth. T. You may write one-tenth on your slates. c. iV T. Very well; I will show you another way to write one- tenth: (.1) one with a period before it. T. You may now write two-tenths in both ways. T. You may now write one and one-tenth in both ways. T. You may now write three and one-tenth. T. You may now write twenty-one and one-tenth. As one-tenth will in practice be used more frequently in the decimal form, it should be given most attention as a decimal form. It should be compared in value with units, as units with tens and tens with hundreds, to clearly show the pupil that it may be treated in all respects as another digit. To that end many ex- ercises like the following should be given: THE SUPPLY. 2X .5 .4 4 40 11 1.1 9 .9 90 7 w .0 3x 7 80 3 .3 12 1.2 8 80 .8 5 .5 9 .9 .4x 3 30 6 60 8 80 70 7 00 9 90 1 10 .t X 5 50 2 20 7 70 4 40 3 30 60 6 80 8 .4 4 40 .7 7 70 .1 1 10 90 9 .9 .8 8 4 .4 1 .1 8 .8 1.2- 1.: 6 .6 4- .4 3 .3 2 .2 1.8-4- Addition: 84.5 35.2 81.9 73.6 48.4 Subtraction : 83. 5 72.9 Division: 5)86.5 Multiplication: 86.4 28 85)95.3( The further consideration of fractional forms may now be pro- ceeded with without the use of either objects or lines, making use of them only in rare cases if at all. The subject of hundredths may profitably be considered while many other fractional forms between tenths and hundredths have received no attention whatever. The reasons for this will be apparent, when it is considered that' hundredths are in constant use, while nineteenths, twenty-thirds, and many other fractional units are rarely used at all. FJL t crroys D AY 'IMA LS. n The pupil is now in the very center of supply and has only to be educated in the subject of demand to make him an adept in the ordinary business affairs of life. The ability to select with promptness, to utilize with ease, to know his needs, to husband his resources; these are yet-to-be-acquired forces that can only be the outgrowth of actual affairs or of wise leading on the part of a thoroughly skilled teacher who is able to send into, and draw from, the business world what is needed, using as his agents the very material into whose hands he wishes to place these new forces. The matter of pointing off in multiplication and division of decimals needs to be dealt with rigidly at this time, so that the applications of percentage will not abound in difficulties not belonging to percentage at all but to the subject of decimals altogether. A few illustrations by means of changing from the decimal to the common form and care thereafter will suffice. 78 THE SUPPLY. DECIMALS PER CENT. When the subject of decimals is thoroughly understood, the idea of per cent may be added with little difficulty. The subject of hundredths should be reviewed and the subject of per cent introduced in some such way as this: T. What have I written? (.02.) C. Two-hundredths. T. It is also called two per cent; per cent meaning hun- dredths. T. You may write three per cent. C. .03. T. Four per cent; five per cent: twelve per cent; eighty-five per cent; one-hundred-eighteen per cent. C. .04; .05; .12; .85; 1.18. T. These numbers are written in another way also: 4%; 5%; 12%; 85%; 118%. T. You may write each of these in three ways, and notice the change in the location of the decimal point. 12% ==.12=^ or ^. 85% =.85=jfo or tf. 118% =1.18 =^=1^ brl-fo. T. Write one-tenth per cent in like manner. C. .1%=.001= T ^. T. How about the location of the decimal point? C. The decimal point is two places farther to the left in the decimal form than it is in the per cent form. T. This is always so. DECIMALS PER CEXT 79 Much practice in the writing of equivalents is an essential preface to a ready application of the principles of percentage. The order of the columns should be changed so that perfect facility in the writing of equivalents may be assured. The columns should also be written singly and read in the different ways, until no hesitation whatever is to be noticed. Then much oral work of the following nature should be given. What is one one-hundredth of one hundred? What is one per cent of one hundred? What is two one-hundredths of one hundred? What is two per cent of one hundred? What is two one-hundredths of two hundred? What is two per cent of two hundred? What is three per cent of two hundred? What is four per cent of two hundred? What is five per cent of two hundred? What is five per cent of three hundred? What is six per cent of three hundred? One is one per cent of what number? Two is one per cent of what number? Two is two per cent of what number? Short and clear explanations should accompany all work in percentage. The following may be considered as fair; though care must be exercised that the pupils do not commit a form to memory without having" grasped its meaning: What is five per cent of three hundred? One per cent of three hundred is three, and five per cent of three hundred is five times three, or fifteen. Or Five per cent equals one-twentieth; and one-twentieth of three hundred is fifteen. Two is two per cent of what number? Since two is two per cent of a number, one per cent of the same number is one, and one hundred per cent of the number is one hundred times one, or one hundred. Or Since two is two per cent, or one-fiftieth, of a number, fifty- fiftieths of the same number is fifty times two, or one hundred. Two is what per cent of five? Two is two-fifths of five; two-fifths of five equals forty hun- dredths, or forty per cent, of five; hence two is forty per cent of five. SO THE SUPPLY. After a clear idea of per cent is gained, the tables ol compound numbers should be reviewed, formulated, and completed, so that the business applications may present as few difficulties as pos- sible. A general review of such points as were troublesome, and of points whose especial needs are recognized, should also be made now and as frequently as seems necessary. Frequent reviews of carefully selected points should be considered not only as proper but as absolutely indispensable. TABULTAIOXS. TABULATIONS. "Exercise involves repetition, which, as regards bodily actions, ends in habits of action and, as regards impressions received by the mind, ends in clearness of perception." Payne. In developing the ideas of the simple numbers the disjointed facts of the tables are incidentally developed, so -that it only remains to systematize these facts and introduce applications in which the ideas of trade are the central ones. MONEY. T. How many cents in a dime ? C. There are ten cents in a dime. T. How many dimes in a dollar? C. There are ten dimes in a dollar. T. One-tenth of a cent is called a mill; then how many mills are there in a cent ? C. There are ten mills in a cent. T. I will now put these facts together so that you may see the table which you are to remember: 10 mills =1 cent. 10 cents =1 dime. 10 dimes 1 dollar. The term "eagle," not being upon any of the coins and rarely used, should not be inserted in the table. The mill, though not a coin, is in constant use in the reading of numbers and in com- putations, and hence should be inserted. Cent, dime, and dollar are upon the coins themselves. si- THE 8V PPL Y. CAPACITY. T. How many pints in a quart? C. There are two pints in a quart. T. How many quarts in a gallon ? C. There are four quarts in a gallon. T. How many gallons in a barrel? C. There are thirty-one and one-half gallons in a barrel, T. How many barrels in a hogshead? C. There are two barrels in a hogshead. T. These facts I will write for you in a table, LIQUID MEASURE. 2 pints =1 quart. 4 quarts =1 gallon, #l-j gallons = 1 barrel. 2 barrels--! hogshead, T. Anna, you may fetch the pint measure,' Jennie, you may fetch the quart measure; and Hattie, you may fetch the gallon measure, T. Class, what do people measure with these measures ? C. Water, sirup, milk, vinegar, wine, honey, beer, and all other liquids, T. With what do people measure wheat? C. People measure wheat with peek, half-bushel, and bushel measures. T. How many quarts in a peck ? C. There are eight quarts in a peck, T. How many pecks in a bushel? C. There are four pecks in a bushel. T. I will write these facts also in a table, DRY MEASURE. 2 pints =1 quart. 8 quarts 1 peck. 4 pecks 1 bushel. T, Lily, you may fetch the quart measure J Charles, you may TABULATIONS. 88 fetch the peck measure; and Samuel, you may fetch the bushel measure. T. Lily, is your quart measure of the same capacity as the one that Jennie brought us? L. My quart measure is somewhat larger than the one Jennie brought. T. What do people measure with these measures ? C. Grain, vegetables, and all other dry or bulky articles that are bought and sold by the quart, peck, or bushel. T. Are these measures used as generally on the Pacific coast as on the Atlantic coast? C. No; grain, vegetables, etc., are generally bought and sold by weight on the Pacific coast. WEIGHT. T. How many ounces in a pound ? C. There are sixteen ounces in a pound. T. How many pounds in a hundred-weight? C. There are one hundred pounds in a hundred-weight. T. How many hundred-weight in a ton? C. There are twenty hundred-weight in a ton. T. These facts may now be written: AVOIRDUPOIS WEIGHT. 16 ounces =1 pound. 100 pounds =1 hundred- weight. 20 hundred-weight = 1 ton. T. Lillian, you may fetch the ounce weight; and Clara, you may fetch the pound weight. T. What articles are bought and sold with these weights ? C. Sugar, meat, hay, grain, live-stock, lime, flour, iron, copper, coal, and all other ordinary articles of merchandise that are bought and sold by weight. T. Do we use these weights in buying and selling gold or silver ? C. We do not. T. Levi, you may fetch me the ounce weight with which people weigh gold and silver. 84 THE SUPPLY. T. Is this ounce weight heavier or lighter, Lillian, than the ounce weight you have? L. It is heavier. T. How many of the ounces that Levi has make a pound with which to weigh gold and silver? C. Twelve ounces. T. Cora, you may fetch the pound weight of which we are now talking. T. Does this pound weight weigh more, or less, than the pound weight with which we weigh sugar? C. It weighs less, T. Then the ounce with which we weigh gold and silver weighs more, and the pound weighs less, does it, than the cor- responding weight with which we weigh sugar and coffee? C. It does, T. The table is: TROY WEIGHT. 24 grains =1 pennyweight, 20 pennyweights 1 ounce. 12 ounces 1 pound, T. These weights are used in weighing gold, silver, platinum, and all jewels except diamonds and pearls, T. After school you may go with me to the drug store of Deveny & Crew, and we will learn whether they use any of these weights, and whether they also use others of which we have none. This excursion may be made of much interest as well as pleasure, as all novel methods of procedure excite at least tem- porary interest, and, if well planned and systematically carried out, the lessons learned should be of the most durable character. These excursions being after school, the pupils should be as free from restraint as possible; in fact, if the teacher is th,e leader he or she should be, no act of any pupil is likely to occur that Will, in any way, mar the pleasure of the occasion. These little outings may be frequently indulged in, if there be a definite object in view, and will therefore be a profitable as well as pleasant feature of the school. The interview with Deveny & Crew should be as informal as possible. These gentlemen will take great pleasure TABULATIONS. so (business men always do) in telling the pupils all about the weighing, in showing them the weights, and in weighing for them different articles and comparing their weights. The in- formation thus obtained will need to be formulated the next day in the classroom, but the impression made will be lasting. T. (The day following the excursion.) Gertrude, you may tell what you learned about weight from Deveny & Crew last evening. G. I learned that in mixing, or compounding, medicines they use weights that resemble the Troy weights somewhat. Their grain, ounce, and pound are exactly the same as the grain, ounce, and pound of the Troy weight. They use the scruple and the dram instead of the pennyweight. Their table is as follows; APOTHECARIES' WEIGHT. 20 grains =1 scruple. 8 scruples 1 dram. S drams =1 ounce. 12 ounces 1 pound. T. That is right. Ralph, you may tell me what else you learned about weight last evening. R. I learned also that they sell by Avoirdupois weight such goods as are sold by weight, and that for liquids they have measures of capacity different from any we have ever talked about in class, T. You may tell us what you learned about them. R. I learned that the table differs from the ordinary liquid measure in having subdivisions, of the pint. The table is: APOTHECARIES' LIQUID MEASURE. tiO minims = 1 dram. ounces 1 pint. And the measures are graduated glass vessels in which very small quantities may be accurately measured. These measures are used both for mixing, or compounding, medicines and in detailing the same when in liquid form. 86 THE SUPPLY. The following will indicate the character of the work necessary to impress the tables upon the pupils' minds: Reduce twenty-five pounds and eight ounces, Avoirdupois weight, to the decimal of a hundred-weight. In forty pounds and eight ounces, Avoirdupois weight, how many pounds, Troy weight? At one dollar and thirty cents per cental, how many tons of wheat can I buy for two thousand six hundred dollars? I bought 675 lb. of hay at $14 per ton ; what did it cost me ? I have 47 Ib. of silver, Avoirdupois weight; what is it worth at SI per ounce Troy? At two per cent more per ounce? TIME. T. How many seconds in a minute? C. There are sixty seconds in a minute. T. You may take my watch, Fred, and show the class the time measured by a minute. T. I will say "one" at the beginning of a minute and "one" again at its end. T. How many minutes in an hour? C. There are sixty minutes in an hour. T. How many hours in a day? C. There are twenty-four hours in a day. T. What is a day ? *C. A day is the time required for the earth to revolve once upon its axis. T. When does the day begin ? C. At midnight. T. When does it end? C. At the next midnight. T. How many days in a week ? C. There are seven days in a week. T. How many days in an ordinary year? C. Three hundred sixty-five days. T. How many days in a leap year? C. Three hundred sixty-six days. T. What is a year? *This answer is not strictly correct hut the error cannot be pointed out at this time. C. A year, is the time required for the earth to revolve around the sun once. T. Yes; the earth revolves around the sun in 365 days, 5 hours, 48 minutes, and 4(5.4 seconds. For convenience, 365 days are called a year. This leaves an excess of 5 hours, 4S minutes, and 46.4 seconds, arid, in four years, an excess of four times 5 hours, 48 minutes, and 46.4 seconds, or an excess of 23 hours, 15 minutes, and 5.0 seconds, very nearly a day; there- fore ordinarily each year divisible by four is given 366 days. This however causes a deficiency of 44 minutes and 54.4 seconds, and in a century, or twenty-five times four years, it causes a deficiency of twenty-five times 44 minutes and 54.4 seconds, or a deficiency of 18 hours, 42 minutes, and 40 seconds; hence the century year ordinarily is given only 305 days. This then produces an excess of 5 hours, 1 7 minutes, and 20 seconds every century, or an excess of four times 5 hours, 17 minutes, and 20 seconds, or an excess of 21 hours, 1) minutes, and 20 seconds, in four centuries; hence every century year divisible by 400 is given 360 days. This again produces a deficiency of 2 hours, 50 minutes, and 40 seconds, and in four thousand years, or ten times four hundred years, it produces a deficiency of ten times 2 hours, 50 minutes, and 40 seconds, or a deficiency of 28 hours, 26 minutes, and 40 seconds: hence 4000 will probably be given only 365 days. Thus leap years are years, except century- years, divisible by four, and century years, except multiples of 4000, divisible by 400. TIME TABLE. <>0 seconds 1 minute. 60 minutes = 1 hour. 24 hours 1 day. 7 days =1 week. 30 days 1 business month. 12 months =1 year. 365 days =1 year. 366 days 1 leap year. TESTS. Reduce 7 weeks, 4 days, and 4 hours, to the fraction of a year. To the decimal of a year. What per cent of a year? ss niK fir P PLY. What time elapsed from January 1, 1850, to July 1(>, 1872? How many days are there in each of the months? How many days are there from January 14 to May 2H? How many days are reckoned as a month, in computing interest? How many weeks are there in a year? Six months are what per cent of a year? Three months ? Four months? Two months? Eight months? Nine months? Ten months ? One month ? DISTANCE. T. How many inches in a foot ? C. There are twelve inches in a foot. T. John, you may draw a line upon the blackboard one foot in length, and divide it into inches. T. You may now take the foot-rule and see how accurately you have done your work. T. How many feet in a yard ? C. There are three feet in a yard. T. Walter, you may draw a line one yard in length, and divide it into feet. T. You may now test your work with the foot-rule. T. How many yards in a rod ? C. There are five and one-half yards in a rod. T. How many rods in a mile? C. There are three hundred and twenty rods in a mile. T. You may now each think of two houses or two places or two objects of any kind that are one mile apart. TABLE OF DISTANCES. VI inches =1 foot. 3 feet =1 yard. 5| yards =1 rod. 320 rods =1 mile. T. George, I shall write a note to County Surveyor Atherton, requesting him to lend us his Surveyors' chain at such time as shall be most convenient for him, and I will thank you to deliver TABULATIONS. ^ the note immediately after school, as County officers close their offices at five o'clock. T. (On the following day, George having brought the Sur- veyors' chain.) Benjamin, you may measure the length of this chain. B. It is exactly 66 feet long. T. You may now measure the distance from one small link to the corresponding next small link. B. It lacks a little of being eight inches. T. Yes; it is exactly 7.92 inches and that distance is called a link. There are one hundred of these in the chain. T. 66 feet equal how many rods? C. 66 feet equal 4 rods. T. This chain is used by land surveyors. SURVEYORS' MEASURE. 7.92 inches = 1 link. 25 links =1 rod. 4 rods =1 chain. 80 chains =1 mile. TESTS. Reduce 84,683 inches to integral higher denominations. From 4 miles, subtract 2 miles, 25 rods, 2 yards, 2 feet, and 9 inches, and find twenty-five per cent of the remainder. Divide 85 miles, 73 rods, 4 yards, 2 feet, and 8 inches into 18 equal parts. Reduce 3 miles, 2 rods, 4 yards, and 8 inches to inches. To links. A and B start at the same time to run toward each other, when they are one mile apart; A runs three-fourths as fast as B; how far does each run before they meet? A room is 40 feet long and 35 feet wide; what will be the expense of carpeting it with carpet f of a yard wide, the strips running lengthwise of the room, there being no loss in matching, and the price being $2. 25 per yard ? How many inches higher is a horse that measures 16^ hands than one that measures 14f hands ? JM THE SUPPLY. How many boards of the longest possible equal lengths will inclose a rectangular field, 9,893 feet long and 8,047 feet wide,- with a straight fence, six boards high? What will it cost, at two dollars per yard, to carpet a room, 20 feet long and 18 feet wide, with carpet three-fourths of a yard wide, the design being one yard in length, and the. strips to run so as to make as little expense as possible. SQUARE MEASURE. The table of distances, or Linear measure, having been thor- oughly taught in connection with the simple numbers, the ideas of squares and square units should be systematically presented in some such way as this: T. James, you may draw a square upon the blackboard, J- T. Samuel, you may draw a line one foot long. S. T. James, you may use Samuel's foot-line and upon it, as one side, draw a square. T. Samuel, you may tell me the name of this square. S. It is a square foot. T. Then what is a square foot, Mary? M, A square foot is a square each of whose sides is one foot long. TABULTAION8. T. Minnie, you may divide each of the sides into inches, using the foot-rule. M. T. How many inches in a foot? C. There are 12 inches in a foot. T. Alice, you may draw parallel lines connecting the points on the opposite sides. A. T. C. T. C. T. C. T. Class, what kind of squares are these small squares ? They are square inches. How many square inches are there in each row ? There are 12 square inches in each row. How many rows are there ? There are 12 rows. Then how many square inches are there in the large square ? C. There are 12 times 12 square inches, or 144 square inches, in the large square. T. What is the large square? C. It is a square foot. T. Then how many square inches are there in one square foot? THE 8 UP PL Y. C. There are 144 square inches in one square foot. T. I will now write this where we may see it all the time, T. Herbert, you may draw a line one yard in length. H. T. How many feet in a yard? C. There are 3 feet in one yard, T. Susan, you may draw a square upon Herbert's line. S, T. What is the name of this, square ? S. A square yard. T. Mortimer, you may divide each side into feet, M. T. Lizzie, you may draw lines parallel to the sides connect- ing these points of division. L, TABULATIONS 1 J3 T. What is each of these smaller squares? C. A square foot. T. How many smaller squares in the large square ? C. There are nine smaller squares in the large square. T Then how many square feet are there in one square yard ? C. There are 9 square feet in one square yard. T. I will place this fact under the fact we learned a little while ago. T. How many yards in a rod ? C. There are 5^- yards in a rod. T. Mabel, you may draw a line one rod in length. M. T. You may now draw a square upon this line as one side. M. The board is not large enough. T. What shall we do then ? M. We may use a shorter line and call it a rod. T. Yes; you may use two inches for a yard. Then how many two-inches will you need for a rad? M. I shall need 5 two-inches and 1 one-inch, or eleven inches. T. That is right; you may now draw your line which we are to use as a rod. M. T. Kate, you may draw a square upon this line as one side, divide the sides into yards, and draw the parallel lines. K. T. Are these divisions all squares? C. No; those on the right and those at the bottom are half- squares. THE SUPPLY. T. What is the one in the corner? C. It is the half of a half-square, or a fourth of a square. T. That is right; let us see how many square yards we have. C. We have 25 square yards. T. How many half square yards? C. JO half square yards, or five square yards more. T. Then how many square yards are there in the square rod ? C. There are 25 square yards and 5 square yards and \ of a square yard, or 30^ square yards. T. We will now write this also with our other facts and com- plete the table: SQUARE MEASURE. 144 square inches 9 square feet ; 30^ square yards 160 square rods 640 acres square foot. square yard. = 1 square rod. 1 acre. 1 square mile, or 1 section. T. Avis, you may draw a figure 3 inches long and 2 inches wide, divide its sides into inches, and draw parallel lines. A. T. How many square inches in a rectangle 3 inches long and 2 inches wide? A. There are three times two, or 6, square inches. Similar simple problems may be given until the pupils are able to state the process for solving problems of that character. Blackboard forms of solution, accompanied by good explanations, should follow, until the subject is firmly fixed in the minds of the pupils: Problem: Measure this room and determine how many square feet there are in the floor. Solution; TABULATIONS. 45 square feet. 36 225 135 1575 square feet. 45 feet. I .")"."> S(|ll;tlV !<> \> ^ fc s % Os OQ Oo~ o s s C. D. having sent word that he will pay the note on Decem- ber 1, 1892, how much will be due and unpaid at that time? Solution: $825 1 year. .08 66.00 825. $891. 1 year, 4 months, 2 days. PERCENT A GE INTEREST. $791 2 years, 2 months, 4 days. .147^ 2 years, 5 months, 17 days. .12 .025 659-J 5537 3164 791 .147 38.978 $155.91 791. $946.91 355. $591.91 .0171 9865 414337 59191 $10.16112 3.38704 12 1 18 3 months 1.3 days. J.oo 591.91 Explanation: This note being of the usual form comes within the purview of the United States rule. The interest from date to the time of the first payment is $66. Since $100 is the pay- ment, the principal will be reduced by such an amount as is not required to pay the interest. The sum of the original principal and the accrued interest, diminished by the payment, produces the true amount of the debt that remains unpaid, or $791. The time intervening to the next payment is 1 year, 4 months and 2 days. It is evident, by inspection, that at 8 per cent the interest will be more than 10 per cent of the principal, and, therefore, more than the payment; as the principal must not be increased , and as the entire payment, and more, is required to pay the accrued interest, it is necessary to compute the interest until such time as the payments are sufficient to more than pay the accrued interest. Since the next interval varies but slightly from the one just 1-20 THE DEMAND. discussed, and since the payment is less than half as great, it is evident that to compute to the time of the third payment would also be useless. The fourth payment is of sufficient magnitude to dispel all doubts, and to make it certain that the aggregate of the second, third, and fourth payments will much more than .pay the interest from the time of the first payment to the time of the fourth payment. The interest from the time of the first payment to the time of the fourth payment is $155.91; this added to the $791, that remained unpaid, produces $946.91. The sum of the last three payments, $355, subtracted from $946.91, leaves $591.91. This last amount is the sum that will draw interest for the remaining time, 3 months and 13 days, to the time of settle- ment. The interest for 3 months and 13 days is $13.55; the sum C. D. must pay, therefore, on December 1st, 1892, is $605.46. A TEST. Southworth & Grattan bought of Hedges & Buck goods to the amount of $1,000, payable in 6 months, without interest. One month afterward, they sold the goods for cash at an advance of 10 per cent, and immediately put the money at interest at 6 per cent. When the 6 months had expired, they collected the amount of the money they had lent, and paid the bill due Hedges & Buck; how much did they gain? BANKING. "I am convinced that the method of teaching which ap- proaches most nearly to the method of investigation is incom- parably j the best. ' ' Biirke. Under the comprehensive term of Banking, the entire business of banks should be considered, so that separate investigations of the subjects of foreign and domestic exchange will be unneces- sary. The subject of Banking will require much skillful direc- tion, great patience, ample time, and good judgment, in order that the results shall be clearly satisfactory. How the vast business of the world is transacted with only an infinitesimal portion of its value in money ever moving at all; who keep the accounts that enable all the settlements to be made at stated PERCENTAGE BANKING. 121 periods with such wonderful precision; what becomes of the coin that is shipped from one nation to another as the balance of trade; how coin that is steadily pouring- in certain directions is replaced: these are living questions, the solution of which raises the student from a lower to a higher plane, and kindles within him new desires and aspirations. He sees the wonders of civilization and the possibilities of the human mind, and is lead to look with awe and reverence to the power that has fashioned it. In Banking, questions of the following nature will lead the pupil to investigate with more skill, having definite points around which to associate his data: Bank Notes: How are they written ? How many signatures are usually required? What is the usual rate of interest? At maturity, what action is taken ? What is the form of a protest? Bank Discount: What is the nature of the notes which banks will purchase? If they do not bear interest, upon what is the discount computed? If they bear interest, upon what is the discount computed ? Deposits: What are "call" deposits? What kinds of deposits draw interest? How does the rate of interest on deposits compare with the legal rate? What is the difference between National Banks, ordinary Commercial Banks, and Savings Banks, as regards deposits ? Drafts and Bills of Exchange: What is the difference between Drafts and Bills of Exchange ? For what purpose are they used? What do we mean when we say ' ' sight drafts ' ' ? Time drafts? What are the conditions of trade that cause drafts or bills of exchange to be at a premium ? At a discount ? Why are two or more copies of Bills of Exchange given ? Do Express and Telegraph Companies do any of these kinds of business? 122 THE DEMAND. CLASS WORK. Problem : At 6 per cent per year, what is the difference be- tween the bank discount and the true discount of a note for $2,059.40, payable in 60 days, without interest? Solution : $20.594 Bank Discount. $2059.40 $20.39 1. 01)2059. 40($2039. 01 pres. value. .20 difference. 202 $20.39 true dis. 394 303 910 909 100 Explanation: Banks, for the purposes of discount, reckon inter- est on the value of the note at maturity. Therefore, the Bank Discount is one per cent of $2059.40, or $20.594. True Discount is the difference between the present value of a note and its value at maturity, and is based on the ordinary method of com- puting interest. The present value, therefore, is such a sum as being placed at interest at the given rate and for the given time will amount to the value of the note at maturity. Since one dollar in 60 days at 6 per cent will amount to $1.01, and since $2059.40 is the amount of the present value for the same time and at the same rate, the present value must be as many times $1 as $1.01 is contained times in $2059.40, or $2039.01; and the true discount is the difference between $2059.40 and $2039.01, or $20.39; and the difference between the Bank Discount and the True Discount is the difference between $20.59 and $20.39, or $.20, approximately. This difference is the interest on the True Discount for the given time and at the given rate. TESTS. Sidney Newell bought 18 shares of bank stock at a premium of 8 per cent on their par value of $100 per share. Six months afterward, and at the end of every subsequent six months, he received a dividend of four and one-half per cent. At the end of two years and three months, he sold the stock at a premium PERCENTAGE BANKING. 123 of 12 per cent. Money being worth 8 per cent per year, com- pound interest, how much did he gain? Having 75 shares of bank stock, I sell it through a broker at 76, paying 1% brokerage. With the net proceeds, I buy a draft at 90 days, on Denver. If sight drafts on Denver cost 101, and if interest is 7%, what is the face of the draft? I paid $2400 for a 60-day draft; exchange being at a premium of l-|-%, and the rate of interest being 8%, what was the face of the draft? A pound sterling being worth $4.8665 in U. S. Gold Coin, how many dollars will purchase a set of exchange to liquidate a debt of ;200 in Liverpool, exchange selling in San Francisco at 3% premium? 124 THE DEMAND. LONGITUDE AND TIME. "Our age inclines at present to the superstition that man is able, by means of simple sense- perception, to attain a knowl- edge of the essence of things, and thereby dispense with the trouble of thinking." Rosencranz. The object of the study of this subject is to gain a clear con- ception of the following facts: (1) That there are two days of the week in existence at the same time, and only two; (2) That the boundary lines of these days is the one-hundred- eightieth meridian,* and the meridian containing the midnight line; (3) That the lune that has the one day is constantly diminishing in width, until its width is nothing, while the lune containing the succeeding day of the week is ' constantly increas- ing in width, until its width is co-extensive with the circumference of the earth; (4) That, at that instant, there is but one day of the week throughout the surface of the earth; (5) That the earth revolves from the west toward the east; (6) That west and east are only relative terms, not absolute directions; (7) That, therefore, when a person looks northward, the surface of the earth moves toward his right, but, if he look southward, it moves toward his left; (8) That a day is that period of time that elapses from the time a given meridian is vertically shone upon by the sun to the time when the same meridian is vertically shone upon the next time; (9) That an hour is one of the 24 equal parts of a day ; (10) That every circle is divided into 360 equal parts called degrees; (11) That, therefore, degrees vary in length as the circumference of the circle varies in length. These facts being definitely established, the applications of the This meridian is the " International Day-line." LONGITUDE AND TIME. 125 principles will not carry with them the meaningless jargon of words that tend to falsify in the minds of students the true idea of education: that the truth fully revealed is the end in view. As valuable aids to the proper conception of this subject, it will be found that the following or kindred diagrams will be exceedingly exhaustive, after the general ideas have been developed with a globe: 90-W '0E.or\V Midnight- 90 F. Noon. 126 THE DEMAND. Noon. Midnight. Noon V SO F. or \V. Greenwich- Midnight- LONGITUDE AND TIME. 127 Noon. Noon. *frdnight- 128 THE DEMAND. oo* wL 90EL '** E. Midnight. The figures should all be drawn on a blackboard on the north wall of the school-room, so that the students in looking at them will have, as nearly as may be, a truthful view of the relative directions of east and west, and of the relative locations of the different portions of the earth. These seven figures are really the same figure, representing different divisions of time. They are each a section of the earth whose boundary is a parallel of latitude, as all parallels are alike as regards time of day, expressed in hours and minutes; when conclusions are drawn with regard to places on one parallel, they will be conclusions regarding the entire surface of the earth. In figure 1, the earth is represented as being in such a position that the sun shines directly upon the meridian which passes through Greenwich. That is the figure with which to open the subject, and the one to dwell upon until by dextrous questioning the following facts are drawn out: (1) That east, to two different persons living on the opposite sides of the earth, is directly opposite directions, and that east at the same place is at no two consecutive minutes the same absolute direction; (2) That a place, 179 degrees, 59 minutes, and 59 seconds east, has nearly midnight of Wednesday afternoon, while one, 179 degrees, 59 minutes, and 59 seconds west, only a few rods distant, has only the very beginning of Wednesday; in other words, the difference in time between two such places, only LONGITUDE AND TIME. 129 a few rods apart, is but an instant less than 24 hours. Let these two thoughts be dwelt upon until no doubt remains. Then, by skillful questioning, establish the fact that while this later time is in east longitude the place that possesses it is west of the one- hundred-eightieth meridian. This is a vital point; for it is necessary that all should fully comprehend this fact in order to understand the change in the calendar that is made by ships in crossing and recrossing the one-hundred-eightieth meridian. That fact being clearly established, figure 2 is ready to be studied as representing the position of the earth at a time sub- sequent to that represented by figure 1. Again, by skillful ques- tioning, lead the pupil to see that if the place on the west side of the one-hundred-eightieth meridian had nearly midnight, Wednes- day, then, when the earth had revolved a little further, the same place must necessarily have Thursday; hence, that the new day always begins on the west side of the one-hundred-eightieth meridian; and that places on opposite sides of this line as the earth continues to revolve, must continue to have nearly 24 hours difference in time. , Then, by a comprehensive examination of figures 3, 4, 5, 6, and 7, the further fact can be established in the minds of the students, that Wednesday must wane as Thursday waxes, that is, that the portions of the earth that have Wednesday must grow less and less, while those that have Thursday grow more and more, until when one revolution of the earth shall have been completed, Wednesday will have expired on the east side of the one-hundred-eightieth meridian, and Thursday will be the day for all portions of the earth; then, Friday will be born on the west side of the one-hundred-eightieth meridian. These principles are fundamental, and indispensable to a clear conception of this very interesting and useful subject. CLASS WORK. Problem : The Longitude of A is 125 degrees west, and of B is 87 degrees and 30 minutes east; what is the time at B when it is 2 o'clock and 20 minutes p. .M. at A? 130 THE DEMAND. Solution : 125 C 87 C 30' 15)212 3(X 14 hr. 10 min. 2 hr. 30 min. 16 hr. 40 min. 4 hr. 40 min. A. M., next day Noon. 87 30' E. A 2 hr. 30 min. P.M. 125 W. Greenwich. Explanation: Since A is 125 degrees west of the meridian which passes through Greenwich, and since B is 87 degrees and 30 minutes east of the same meridian, it is evident that B has revolved through the sum of 125 degrees and 87 degrees and 30 minutes, or 212 degrees and 30 minutes, since it had the same time that A now has. The earth revolves 15 degrees in one hour; to revolve 210 degrees will require 14 hours; to revolve 2 degrees and 30 minutes, or 150 minutes, will require 10 minutes of time, since the earth revolves 15 minutes of longitude in one minute of time. Therefore, it is 14 hours and 10 minutes since B had the same time that A now has; hence the time that B has is 16 hours and 40 minutes past noon, or 4 hours and 40 minutes A. M., the next day. Problem: It is 8 hours and 40 minutes A. M. at A when it is 1 hour and 20 minutes A. M. at B, whose longitude is 25 degrees and 30 minutes east; what is the longitude of A? LONGITUDE AND TIME. 131 Solution: 8 hours 40 minutes. 1 hour 20 minutes. 7 hours 20 minutes. 15 110 25 30* 135 30' east. Noon. S hr. 40 min. A. M. (?) A L hr. 20 min. A. M. 25 3(X E. Greenwich. Explanation: Since the time at A is 8 hours and 40 minutes, and of B 1 hour and 30 minutes, both A. M. , of the same day, A must have had the same time that B now has 7 hours and 20 minutes before; and, since the earth revolves to the eastward, A must lie east of B. The earth revolves 15 minutes of longitude in one minute of time; in 20 minutes, therefore, it revolves 20 times 15 minutes of longitude, or 300 minutes of longitude, or 5 degrees. It revolves 15 degrees in 1 hour; in 7 hours, it revolves 7 times 15 degrees, or 105 degrees; hence, in 7 hours and 30 minutes, it revolves 105 degrees plus 5 degrees, or 110 degrees; hence A lies 110 degrees east of B, or 135 degrees and 30 minutes east of the meridian which passes through Greenwich. TESTS. What are the boundary lines between the days? Where does the day begin ? Where does the day end ? 132 THE DEMAND. What is the position of the earth with regard to the sun when there is but one day on the surface of the earth? What is the position of the earth when one-half of the earth's surface has one day, and the other half the succeeding day ? What is the position of the earth when one-fourth the earth's surface has one day, and the remaining three-fourths has the suc- ceeding day? What is the position of the earth when one-fourth the earth's surface has one day, and the remaining three-fourths has the preceding day? At what time of day for us does the succeeding day begin at its place of beginning? In what direction does the earth revolve? How far does the earth revolve in one hour? In one minute? In one second? Do places east of us have earlier, or later, time than we? At 2 o'clock A. M. of Monday for us, is it Sunday, or Tuesday, in some other part of the world? At 10 o'clock A. M.? At 3 o'clock p. M.? The longitude of A is 175 degrees and 25 minutes west, and of B is 28 degrees and 40 minutes west; what is the time at B, when it is 3 hours and 25 minutes p. M. at A? The longitude of A is 45 degrees and 30 minutes east; what is the longitude of B, whose time is 6 hours 50 minutes A. M. when it is 8 hours and 20 minutes p. M. at A? THE METRIC SYSTEM. THE METRIC SYSTEM OF WEIGHTS AND MEASURES. "In whatever it is our duty to act, those matters also it is our duty to study." Dr. Arnold. The Metric System of Weights and Measures has been adopted by, and is used in a modified or unmodified form in, France, Spain, Italy, Greece, Holland, Germany, Austria, Nor- way and Sweden, Denmark. Switzerland, Portugal, Belgium, Turkey, Mexico, Central America, Brazil, Ecuador, Peru, Chili, Venezuela, United States of Columbia, Argentine Republic, Uruguay, Paraguay, Haiti, Congo Free Stats, British India, Mauritius, and others; while the United States of America and Great Britain have authorized its use within their respective pos- sessions. Its use within the last two countries is, however, at present, very limited indeed. The merits of the system are recognized, nevertheless, by the citizens of both the United States and Great Britain, or rather, by those who have thorough- ly investigated the subject of exchange. It, therefore, becomes the duty of all who are connected with educational matters and institutions to become masters of this subject, to the end that the schools may become the medium through which this nation shall advance to, and take her place in, the front rank on this subject also. Much discretion is required in the presentation of this subject; for the aim should be not alone to teach the subject, but to impress upon the pupil's mind the beauties and advantages of the system. Hence the custom of changing from the units of one system to those of the other should not be followed at all. All work should be done wholly within the Metric System, or within the 134 THE DEMAND. common system, then the pupil will readily draw correct con- clusions as to the comparative merits of the two. If the subject is properly presented, the student will see that all operations, reduction ascending" and descending, reduction from capacity in liquid units to capacity in solid units, to weight even, under cir- cumscribed circumstances, is a matter simply of removal of the decimal point; that, as a system of weights and measures, when compared with the common system, it is analogous, in its simple decimal units, to our money system, as compared with that of Great Britain, which we inherited, disliked, and wisely laid aside. As the Metric System has generally been taught, if taught at all, (that is, by reducing gallons to liters, steres to cords, miles to meters, and the like) the pupil, and the teacher even, is uncon - sciously led to the false conclusion, that the processes are com- plicated, and, therefore, that the system is a failure. The teacher forgets, if he has ever thought, that the business man will have no occasion to revert to the old system if the new is employed; he forgets that the only changes which will be needed, or ever used, will be changes from meters to kilometers, from liters to centiliters, from decigrams to quintals, and the like; and that all these changes can be made without any other process than that of removing the decimal point. He does not realize that 39.37079+ inches will be perfectly useless to him then; that 35.316+ cubic feet will not be needed; that .908+ quarts, and all the other terrible numbers, will be of less value to him than the hieroglyphics on Cleopatra's needle. Then, let the following tables, substantially as herein given, be all that is taught or learned from the text; and let all processes be wholly within the system, and consist in the removal of the decimal point only, whenever possible; and that is impossible only when computing square and cubical contents, having the linear units given. TABLE OF LENGTHS. 10 Millimeters \ Centimeter. 10 Centimeters =1 Decimeter. 10 Decimeters =1 Meter. 10 Meters 1 Dekameter. 10 Dekameters =1 Hectometer. 10 Hectometers =1 Kilometer. 10 Kilometers 1 Myriameter. THE METRIC SYSTEM. 135 TABLE OF SURFACES. centiare. 100 square millimeters =1 square centimeter. 100 square centimeters =1 square decimeter. 100 square decimeters =1 square meter, or 1 ct 100 square meters =1 square dekameter, or 1 are. 100 square dekameters =1 square hectometer, or 1 hectare. 100 square hectometers =1 square kilometer. 100 square kilometers =1 square myriameter. TABLE OF SOLIDS. 1000 cubic millimeters =1 cubic centimeter. 1000 cubic centimeters =1 cubic decimeter. 1000 cubic decimeters = 1 cubic meter, or 1 stere. 1000 cubic meters 1 cubic dekameter. 1000 cubic dekameters 1 cubic hectometer. 1000 cubic hectometers =1 cubic kilometer. 1000 cubic kilometers =1 cubic myriameter. TABLE OF CAPACITY. 10 milliliters =1 centiliter. 10 centiliters =1 deciliter. 10 deciliters =1 liter. 10 liters =1 dekaliter. 10 dekaliters =1 hectoliter. 10 hectoliters =1 kiloliter. 10 kiloliters =1 myrialiter. TABLE OF WEIGHT. 10 milligrams =1 centigram. 10 centigrams =1 decigram. 10 decigrams =1 gram. 10 grams =1 dekagram. 10 dekagrams =1 hectogram. 10 hectograms =1 kilogram. 10 kilograms =1 myriagram. j.u Kilograms =1 mynagr 10 myriagrams 1 quintal. 10 quintals =1 ton. \m THE DEMAND. EQUIVALENTS. 1 stere=l cubic meter =1 kiloliter =1 ton of pure water. 1 cubic decimeter =1 liter =1 kilogram of pure water. I cubic centimeter = 1 milliliter = l gram of pure water. 1 cubic millimeter =1 milligram of pure water. Actual weights and measures should be in the class room whenever possible. For all practicable purposes, there are but three names of units and seven prefixes to be committed to memory, in all, ten words only. The task of learning the tables is, therefore, a very light one; for the numbers require no attention worthy of mention. THE PUBLIC LANDS. 137 THE PUBLIC LANDS. In the original Thirteen States, the Public Lands belonged originally to the respective states in which they are situated; and they were, by the several states, sold or otherwise disposed of in accordance with the laws of those states. Each state, therefore, has a system of subdivisions of the lands differing more or less from that of each of the other of the original Thirteen States; and, frequently, in different parts of the same state, widely different methods of subdividing the lands obtained. Texas, having been a separate Republic, at the time of her annexation, retained full control of the Public Lands within her boundaries. Her system of subdivisions of the lands is, however, analogous to that of the United States. The states made from portions of the original Thirteen States also retained control of the Public Lands within their respective limits. In all other portions of the United States, the Public Lands belonged to the United States Government; and ownership to the same has been, or must be, acquired under the laws of the United States. The United States Government has established a uniform system of surveys of the Public Lands, which is both simple and complete. It consists in dividing and subdividing the entire domain, by north and south and east and west lines, into town- ships, as nearly as may be, six miles square; and these townships are divided into sections one mile square, or as nearly as may be. The following plats and descriptions will more minutely represent the scope and plan of the system : 138 THE DEMAND. TOWNSHIPS. Stan? lard Pan illel. TWTE5E T5HE4W O) 1 3 T3NB5W T3NE2E Base T1NE1W Ini d T1NE1E tial fe Line. =5? Poi T1SE1W nt. T1SE1E T2SS5E T3SB4W C .5 'S T4S32E Stan dard 2 9 ~ ? \ t\ r. l\ n *7 /^ n <- _ I \ 9 i i \ ? h ^ r / _<) J JL j tJ J Y J J J 6 Sections are one mile square, or one thirty-sixth of a township, as nearly as may be, considering the convergency of meridians, and errors otherwise inseparable from the system of surveys. All errors, excesses, and deficiencies, from whatever cause, are crowded against the northern and western boundaries of the township. Hence, all sections, except those against the northern and western boundaries of the township, contain exactly 640 acres, while those excepted usually contain either less or more than 640 acres. Sections one, two, three, four, five, seven, eighteen, nineteen, thirty, and thirty-one, are so divided that they, severally contain two full quarter-sections (160 acres) and two full eighties; while section six is so divided that it contains one full quarter-section, two full eighties, and one full forty-acre lot. 14(1 THE DEMAND. The remainders of these sections are divided into lots containing somewhat more or less than forty acres, and are accounted to contain neither more nor less than their true acreage. ^ 40 40 40 100 A A ICO A 'i 160 A 160 A 160 A 160 A THE P UBLIC LANDS. 141 Corners are established every half-mile on the exterior bound- aries of townships and sections, as shown in the diagrams, and whenever practicable, they are marked as therein shown. In addition to the marking of the post which is set as shown in the diagrams, one tree at each post and in. each section is marked BT, together with the name of the township and section in which it is situated; and its direction and distance from the corner is recorded in the field-notes. Any tree that is exactly on the line is marked with two notches, or chops, on each of its opposite two sides in the directions of the line. Trees near the line are blazed twice, once facing the direction whence the line was surveyed, and once facing the direction whither it was surveyed, both blazes being nearer the side of the tree facing the line at right angles, than the one opposite the line. The foregoing are the methods employed in surveying in timbered localities. In stony localities, a monument of stones is used for a township corner, and a single stone for a section and a quarter-section corner. All posts and stones used as corners are notched as follows, on the edges facing the cardinal points of the compass : Township corners, six notches on each edge; all other section corners are notched with as many notches on their respective edges as will indicate the number of miles the corner is situated from the township boundary which the edge faces. In portions of the country in which there is neither stone nor timber, mounds of earth are thrown up at the section corners in a conical shape, and, surrounding each mound, a trench in the form of a square, with its corners on the line. The trenches at the half-mile corners are of the same shape, but the sides of the square are parallel with, and p'erpendicular to, the section lines. Mounds of earth or stone are built, also, where there are no bearing trees, even if there are marked posts or stones. Within the mound is deposited a stone or some charcoal. The form of the mounds, trenches, and pits, which are inseparable, are as shown in the annexed diagrams: 142 THE DEMAND. o PIT PIT o o PIT PIT o PIT PIT THE PUBLIC LANDS. 148 TESTS. Draw a plot of a section of land, and locate the N. E. ^, and the N. \ of S. W. ^ Draw a plot of a township of land, and locate the N. W. \ of the S. E. \ of section 9. How many acres are there in the location? Draw a plot of a township of land; subdivide it and locate the S. W. \ of section 11, the S. |- of section 10, and the W. | of the N. W. \ of section 15. What is the land located worth at $25. per acre, and what will it cost to fence it at $1 per rod ? A man in New York City, owning a half section of land, requested me to sell it for him at $65 per acre, on a commission of one and one-half per cent, and to remit the proceeds to him in a 60-day draft; exchange being at 2 per cent premium, and interest 6 per cent, what was the face of the draft? The man who bought the land mentioned in the preceding problem wished me to have it fenced for him at $1.50 per rod, he allowing me a commission of three per cent; and he then wished me to sell the land for him at a net profit of 10 per cent, allowing me a commission of 1 per cent for selling; for how much was the land sold? 144 THE DEMAND. AVERAGE OF ACCOUNTS. This subject is of immediate interest to only a small portion of the people, and therefore should occupy a secondary position. The purpose of averaging accounts is to determine the date from which interest shall be computed on the balance of the account, so that there shall be no loss to either party; or, to ascertain the date of a note given in settlement, and whose face is the balance. This subject requires but little outside investigation, as all the ordinary business transactions leading up to it are already familiar to the pupils. The methods of computing, and the interpretations of the restflts, when the date from which interest is computed falls either before all the dates of the transaction, or after all the dates of the transaction, constitute the important work under this subject. In other words, the interest of the student will center in the methods of procedure and the results. DeGarmo says: "The teacher is needed for the steps which the children cannot take alone, the derivations and applications which they would not or could not make; con- sequently, instruction should deliberately plan for these greater matters of education, leaving the small ones to an awakened spontaneity of the pupil, or to incidental instruction." CLASS WORK. Problem: The following account was settled by the debtor giving his note for the balance. Determine the face, date, and maker of the note. A VERAGE OF ACCO UNTS. 14o Dr. A. B. 1890. May 6 Mdse $7150. 16 " 475. June 17 " 3475.25 21 " 1516.50 July 5 " 279.90 Dr. Solution: $7150. x 0- 0. 475 xlO= 4750. 3475.25x42 = 145960.5 1516.50x46- 69759. 279.90x60= 16794. $12896.65 237263.5 Cr. 1890. May 9 Cash $2450. 21 " 915. June 12 Mdse 4165.50 19 " 2915.50 3475.25 42 Cr. $2450. x 3 = 915 x!5 = 7350. 13725. 4165.50x37= 154123.5 2915.50x44= 128282. $10446. 303480.5 2450.65 ) 66217.00 49013 (27 2915.5 44 17204 00 17154 55 116620 116620 128282 695050 1390100 145960.50 1516.5 46_ 90990 60660 69759 4165.5 37 291585 124965 154123.5 May 6 27 days = April 9 = date of note. $2450. 65= face of note. A. B. =the maker of the note. Explanation: The debit side of the account represents what A B owes to C D, and the credit side of the account represents what C D owes to A B. The debit side being $2450.65 greater than the credit side, the face of the note must be $2450.65 and the maker A B. The date of the note is yet to be determined. If the note were dated May 6, 1890, A B would neither gain nor lose any interest on the $7150, since that amount of merchandise was bought on that day. He would however lose the interest 146 THE DEMAND. of $475 for ten days, or the interest of $1 for 4750 days; he would lose the interest of $3475.25 for * 42 days, or the interest' of $1 for 145,960.5 days; also the interest of $1516.50 for 46 days, or the interest of $1 for 69,759 days; and the interest of $279.90 for 60 days, or the interest of $1 for 16,794 days. He would lose, therefore, the interest of $1 for 237,263.5 days, on the debit items, if the note were dated May 6, 1890. He would, however, gain the interest of $2450 for three days, or the interest of $1 for 7,350 days; the interest of $915 for 15 days, or the interest of $1 for 13,725 days; the interest of $4,165.50 for 37 days, or the interest ol $1 for 154,123.5 days; and the interest of $2915.50 for 44 days, or the interest of $1 for 128,282 days. He would gain, therefore, the interest of $1 for 303,480.5 days, on the debit items; he would gain on the whole account the interest of $1 for 66,217 days; but the interest of $1 for 66,217 days is equal to the interest of $2450.65 for 27 days. Hence the note must be dated 27 days anterior to May 6, or April 9, 1890. PROPORTION. 147 PROPORTION. Since all problems that can be solved arithmetically by pro- portion can be solved quite readily under other principles, it seems that proportion, from an arithmetical stand-point, is more valuable as a device than as a principle. If it is to be considered as a device, it should be employed in such way as to avoid as much mental and physical labor as possible. In other words, it should be as simple and direct as possible. By the nature of business transactions it is evident that cost and quantity, time of labor and labor, etc., under given conditions 'bear the same relations to other like quantities under the same conditions; hence proportion is simply a comparison of the elements of two transactions, in one of which the elements are all known, and in the other of which one element is unknown. The method is clearly outlined in the solutions and explanations of the following problems : Problem: If a garrison of 3600 men have bread enough to last them thirty-five days, if each be allowed 24 ounces per day, how many men, at 14 ounces per day each, will require twice as much for 45 days? Solution: 3600 ? 35 : 45 :: 1 : 2 24 14 400 2 X 24 x -35 x 3600 men =9600 men. Explanation : It is evident that the ratio which 3600 times 35 148 THE DEMAND. times 24 ounces bear to ? times 45 times 14 ounces is as one is to 2. It has already been established that, In any proportion the product of the means equals the product of the extremes. 0600 is the number that will produce this equality; hence 9600 men is the answer to the question. Problem: If 82 men build a wall 36 feet long, 8 feet high, and 4 feet thick, in 4 days, in what time will 48 men build a wall 864 feet long, 6 feet high, and 3 feet wide? Solution: 82 : 48 :: 36 : 864 4 ? 8 6 4 3 9 ft* 41 $X0x804x82x4days = 369=92| days. 4x0x$$x$* 4 4 Explanation: It is evident that the ratio that 82 times 4 days bears to 48 times ? days is equal to the ratio that 36 times 8 times 4 cubic feet, built by the 82 men in 4 days, bears to the 864 times 6 times 3 cubic feet, built by the 48 men in the required number of days; for the quantity of work performed always has a given relationship to the number of days required to perform it, all other conditions being equal. It has already been established that, In any proportion the product of the extremes equals the product of the means. 92^ is the number that will produce this equality; therefore, 92^ days is the time required for the 48 men to work. GENERAL AVERAGE SHIPPIXU. 149 GENERAL AVERAGE SHIPPING. <( The aim of education must be to arouse in the pupil this spiritual and ethical sensitiveness which does not look upon any- thing as merely indifferent, but rather knows how to seize in everything, even in the seemingly unimportant, its universal significance. " Rosencranz. This is one of the subjects that directly affect such a small portion of the population that its treatment should be reserved until the subjects of more nearly general application are thor- oughly comprehended. In seaboard towns and cities, it will be a more vital subject than in inland localities where the shipping interests are a minimum. Hence, its consideration should cover a greater period of time, be much more exhaustive in its nature, and, perhaps, be undertaken at an earlier age by a maritime people than by others. This may be stated as a general prin- ciple, that subjects should receive attention in accordance with the vocations of the people. Students, in starting out to investi- gate the subject of General Average, should be given an outline of points to be their guide, so that random work may be avoided and valuable time saved. . OUTLINE. 1. Interview ship owners, ship captains, and shippers. 2. Ascertain the value of ships and cargoes. 3. Ascertain the time and expenses of voyages. 4. Ascertain the rates of freightage. 5. Review maritime insurance. 6. Ascertain the frequency and extent of disasters. 7. Ascertain the probable duration of ships. 150 THE DEMAND. 8. Ascertain the mode of procedure in cases of imminent peril, 9. Ascertain how losses are sustained; whether by persons immediately affected or by all, and, if by all, in what ratio. Out of the vast amount of material gathered, the skillful teacher can select, classify, and retain, as working power, for the pupils' future guidance, the essentials only. The matter of com- merce however after this investigation is to the pupil no longer a terra incognita, but a discovered and active realm, affecting the well being of the entire civilized world. The possibilities of man are, in the minds of the pupils, greater and greater every day of their lives. They are more and more impressed with the idea that they must do and become more and more as the days pass, if they would attain anything above mediocrity. The sluggard or the dullard, they are convinced, has no place in such an active world. Onward and upward is the only course that will adorn and honor. Teaching is a pleasure when the pupils are imbued with these ideas. Out of abundant desire fruitful results are sure to grow. ALLIGATION. Alligation is another subject whose importance is restricted, owing to the fact that few employ its ordinary methods in any business transactions. When the quantities and price per unit of quantity are given, and the average price, without gain or loss, is required, the process is so simple as to need no explana- tion. The phase of the subject that calls for thoughtful con- sideration is the one in which the average price and the several prices per unit are given, to find sets of quantities that will satisfy these conditions, without gain or loss. Sometimes certain in- gredients are to be used in specified amounts in order to work off a certain kind of stock, and sometimes the entire mixture is to be of a specified quantity in order to fill a certain vessel, a certain order, or to fulfill any other certain specified conditions ; then the problem becomes severer. MENTAL DISCIPLINE, 151 MENTAL DISCIPLINE. "Repeated attacks, by concentrated attention, not only master the abstruse problem, but leave the mind with a permanent acquisition of power of analysis for new problems." Harris. Problem: A can do a piece of work in 8 days, and B can do the same work in 9 days; in what time can both do the same work, working together? Solution: |+i=H- H+tt=& days. Explanation: Since A can do the work in 8 days, in 1 day he can do one-eighth of the work; and since B can do it in 9 days, in one day he can do one-ninth of the work; both, therefore, in one day can do the sum of one-eighth and one-ninth of the work, or seventeen seventy-seconds of the work; and to do seventy-two seventy-seconds, or the whole work, will require as many times one day as seventeen seventy-seconds is contained times in seventy-two seventy-seconds, or four and four-seven- teenths times one day, or four and four-seventeenths days. Problem : A and B can do a piece of work in 14 days, and A can do six-sevenths as much as B; in what time can each do the work alone? Solution : Y of 14 = 30 i da Y s . A - ty of 14 =26 days, B. Explanation: Since A can do six -sevenths as much as B in a given time, both can do thirteen-sevenths as much as B in a 152 THE DEMAND. given time. Since A was 14 days doing six-sevenths as much as B did in 14 days, to do one-seventh as much would require one- sixth of 14 days, and to do thirteen-sevenths as much, or what both did, would require thirteen times one-sixth of 14 days, or 30$ days. Since B was 14 days doing seven-sevenths as much as he did in fourteen days, to do one-seventh as much would require one-seventh of 14 days, or 2 days; and to do thirteen-sevenths as much, or what both did, would require thirteen times 2 days, or 26 days. Problem: A and B can do a piece of work in 8 days; A and C can do the same piece of work in 9 days; and B and C in 6 days. In what time can all do the work, working together? In what time can each do the work alone? Qr1nfir>n 1 _l_ 1 _J_ 1 _ 29 29 . 9__29_ 144 . 29 __ A 28 rlotrc oil on - fTT'TT^rffj TT~^ TTT> T~TT7 4 2~9 days, all. 29 _ 1 . _ 11 144 . 11 _ TT f T4"T' lf~ days, B. -T^=28i days, A. Explanation: Since A and B can do the work in 8 days, in 1 day they can do one-eighth of the work; and likewise A and C in 1 day can do one-ninth of the work, and B and C in 1 day can do one-sixth of the work. Then twice what A, B, and C can do in one day is the sum of one-eighth, one-ninth, and one-sixth of the work, or twenty-nine seventy-seconds of the work; hence what they can do in 1 day is one-half of twenty- nine seventy-seconds, or twenty-nine one-hundred-forty-fourths, of the work. They can do the entire work, therefore, in as many days as twenty-nine one-hundred-forty-fourths is con- tained times in one-hundred-forty-four one-hundred-forty-fourths, or four and twenty-eight twenty-ninths times one day, or four and twenty-eight twenty-ninths days. All can do twenty-nine one hundred-forty-fourths of the work in one day; A and B can do one-eighth of the work in one day; C, therefore, can do the difference between twenty-nine one-hundred-forty-fourths and MENTAL DISCIPLINE. 153 one-eighth, or eleven one-hundred-forty-fourths of the work in one day. He can do the entire work in thirteen and one- eleventh days. Likewise, B can do it in eleven and one-thirteenth days, and A in twenty-eight and four-fifths days. Problem : At what time between two and three are the hands of a clock twenty minute-spaces apart? Solution:. yfx 30=32 T 8 T minutes past 2. and |2. x50=54 1 5 r minutes past 2. Explanation: Since the minute hand passes entirely around the face of the clock while the hour hand passes over a five- minute-space, the minute hand gains on the hour hand eleven minute-spaces in going over twelve minute-spaces. At two o'clock the hands are ten minute-spaces apart; it is evident, therefore, that the minute hand must gain ten minute-spaces to over-take the hour hand, and must gain twenty minute-spaces more to be twenty minute-spaces beyond it. Hence the minute hand must gain in all thirty minute-spaces; and as it gains eleven spaces in going twelve, it will go twelve-elevenths of thirty minutes -spaces after two o'clock. Hence, at thirty-two and eight-elevenths minutes past two o'clock, the hands are twenty minute-spaces apart. They are twenty minute-spaces apart again at fifty-four and six-elevenths minutes past two o'clock. Problem: What is the time when the minute hand lacks as much of being at the IV-mark as the honr hand is beyond it? 154 Solution: THE DEMAND. of 20 =18& past 4. Explanation: The minute hand moves twelve times as fast as the hour hand, hence when the hands have the required position the minute hand will have gone twelve times as far since four o'clock as the hour hand, and since it is then as far back of the IV-mark as the hour hand is beyond it, it is clear that the distance from the XI I -mark to the IV-mark is thirteen times the distance that the hour hand is beyond the IV-mark; the minute hand is only twelve-thirteenths of the distance to the IV-mark; if it were at the IV-mark the time would be twenty minutes past 4 o'clock; the time, therefore, that fulfills the conditions of the problem is twelve-thirteenths of twenty minutes past 4 o'clock, or eighteen and six-thirteenths minutes past 4 o'clock. Problem : I wish to pay both principal and interest of a debt of $4000 in four equal payments at the end of 1, 2, 3, and 4 years, respectively, the rate of interest being 7 per cent. What shall be the amount of each payment? How much of each pay- ment is principal, and how much is interest? Solution: 100% =lst payment of principal. 107% =2d payment of principal. 114.49% =3d payment of principal. 122. 5043%= 4th payment of principal. <<0 .9943% of the 1st payment of principal = the 4 icinal. or #4000 443.9943% of the 1 payments of principal, or $4000. MENTAL DISCIPLINE. loo 443. 9943)4000. 0000(9. 0091 = 1 % . 3995 9487 4 0513000 3 9959487 5535130 4439943 100%=$ 900.91=lst payment of principal. 280. =lst payment of interest. 1180. 91= amount of each payment. 4000 9.0091 900.91 107 3099.09 630637 07 90091 216.9363 $963.9737 2d payment of principal. 216.9363 2d payment of interest. 1180.91 amount of each payment. Explanation: During the first year the entire $4000 draws interest; during the second year, a portion of the principal having been paid, the interest is less than the interest the first year; hence more principal must be paid to keep the entire pay- ments equal. Likewise, each succeeding year the interest de- creases and the payment of principal increases. The payment of principal the first year is one hundred per cent of itself. The second year no interest will be required to be paid on this hundred per cent; but seven per cent of one hundred per cent, or seven per cent of the first payment of principal, is the amount of interest this payment of principal bore the first year; hence, the second year the interest will be less than the interest the first year by seven per cent of the first payment of principal; hence, to make the entire second payment equal the entire first payment, the payment of principal must be increased seven per cent of the first payment of principal; hence the second payment of principal must be one-hundred-seven per cent of the first payment of principal. By the same course of argument, the third payment of principal will be seven per cent of the second payment of principal more than the second payment of principal, and the fourth payment of principal seven 156 THE DEMAND. per cent of the third more than the third. All the payments of principal are, therefore, 443.9943% of the first payment of principal; but all the payments of principal are $4000. Hence 443.9943% of the first payment of principal equals $4000; one per cent equals $9.0091; and one hundred per cent, or the first payment of principal, equals $900.91. The interest on $4000 for one year at seven per cent is $280; hence the entire payment the first year, and consequently every year, is $1080.91. The pay- ment of principal the second year is 107 X $9. 0091; the third year, 114. 49 x $9. 0091; and the fourth year, 122. 5043 x $9. 0091. The interest the second year is seven per cent of $4000 =$900. 91; and that the third and fourth years is similarly ascertained. Problem: A man wishes to pay principal and interest of a debt of $5000 in three equal payments in 0, 1, and 2 years from date. If the debt bear 8 per cent interest, what must be the amount of each of the three equal payments that will discharge the debt ? How much of each payment is principal, and how much is interest ? Solution: 100% =payment of principal at end of 1 year. 108% =payment of principal at end of 2 years. 116.64% = payment of principal at end of years. 324.64% of payment of principal at end of 1st year =$5000. 324.64)5000.00(15.40167=1% of payment of principal at end 3246 4 of 1st year. 1753 60 1623 20 130 400 129 856 219360 194784 24576 100% =$1540.17=2d payment of principal. 108% = 1663. 38= 3d payment of principal. 116.64% 1796.45=cash pay't, or pay't in years. MENTAL DISCIPLINE. 157 5000 1796.45 3203.55 .08 $ 256.284 interest at end of 1st year. 1540.17 $1796.45 entire payment at end of 1st year. Explanation: The payment in years, that is, the cash pay- ment, is necessarily all principal, as no interest has had time to accrue. The second payment, or the payment at the end of the first year, contains the most interest, and, therefore, the ' smallest payment of principal. Then by the same course of argument employed in the preceding problem the second pay- ment of principal will be 100% of itself, and the third payment of principal 108% of the second payment of principal. The third payment of principal is the final payment of principal and, consequently, the amount of interest paid at the final payment is 8 per cent of 108 per cent of the second payment of principal. But 8 per cent of 108 per cent, added to 108 per cent, equals 116.64 per cent, which is the entire amount paid at the last payment. The entire amount paid at the last payment equals the entire amonut paid at any other payment; consequently, the cash payment must be 116.64 per cent of the principal paid at the second payment. Then 324.64 per cent equals $5000; and the payment of principal at the end of the first year (second pay- ment) equals $1540.17; at the end of the second year, (last pay- ment) $1663.38; and the cash payment, (first payment) $1796.45. This payment was all cash, hence the entire payment each year was $1796.45. $5000 $1796.45, or $3203.55, at 8 per cent for one year equals $256.28, the interest paid at the end of the first year. $1540. 17 +$256. 23 =$1796.45, the entire payment again. The princpal to bear interest the second year is the third pay- ment of principal, $1663.38, which at 8 per cent for one year produces $133.07, the last payment of interest. $1664.38+ $133. 07 =$1796. 45 the entire payment again. TESTS. A and B can do a piece of work in 9 days; A and C can do 158 THE DEMAND. the same work in 12 days; and B and C, in 15 days. In what time can each do the piece of work alone ? A and B can do a piece of work in 15 days; A can do five- sixths as much as B in a given time. In what time can each do the piece of work alone? When will the hands of a watch be together between two and three o'clock? Opposite? At right angles? Five minute-spaces apart? Twenty minute-spaces apart? What is the time when the hour hand is as far beyond the Vl-mark as the minute hand lacks of being at the VUI-mark? What is the time when the minute hand is twice as far beyond the II-mark as the hour hand lacks of being at the V-mark ? Divide $2000 among A, B, and C, in the proportion of one- third, one-fourth, and one-sixth. I wish to pay the interest and principal of an interest-bearing debt of $2000, in four equal annual payments, one, two, three, and four years from date. The rate of interest which the debt bears is six per cent. How much must I pay at each payment ? How much of the first payment is principal, and how much is interest? Of the fourth? I wish to pay the interest and principal of an interest-bearing debt of $4000, in four equal payments, one year apart; the first in cash, the second at the end of one year, the third at the end of two years, and the fourth at the end of three years. The rate of interest which the debt bears is 8 per cent. How much must I pay at each payment? How much of the second payment is principal, and how much is interest ? Of the third ? A, B, and C can do a piece of work in nine days; A can do three-fourths as much as B; and B can do four-fifths as much as C. In how many days can each do the work alone? RESUME. 159 RESUME. "The entire succession of men, through the whole course of ages, must be regarded as one man always living and constantly learning. " Pascal. In considering the grandeur of this nation, the people are considering the potency of the public school system; for the march of empire has been along the pathway made by the public-school teacher. He has been the vanguard of progress, the center of civilization, the harbinger of victory, the mainstay of liberty. His advent has always been hailed with pleasure, and his presence has been a source of endless prosperity, He has toiled for the people for centuries in a silence that has been golden for them, while he has received, in this world's goods, little reward. The public-school system from its humble origin has followed the silent school teacher, and as silently done its great work. It has drawn from the firm granites among which it started, the very elements of strength and endurance. Like the sturdy oak, it has extended its branches over a broad area, and been a source of admiration and joy for generations. Like the spreading banyan, its branches have taken root in other soils, without severing their connection with the parent stock, thus drawing new life and increased vigor from the newer soils, while not losing the benefit of the support of the central trunk that has been its honor and glory. It is a grand idea, this confederation of states in the educational union. In no other functions are the American people so thoroughly a unit, as in educational thought. They may be sectional and industrial as regards tariff and free trade, partisan as regards state's rights and centralization of power, sectarian as regards religion; but 160 RESUME. i as regards education, they are only patriotic. In short, the educational system stands out, what it is, the greatest American institution, full of eloquent possibilities; so grand and so inti- mately a part of our very national and social greatness, that it seems almost a reproach that in the Cabinet of the President of the United States there sits no one whose special duty it is to represent the educational interests of this people. There should be a Secretary of Education to sit the peer of the Secretary of War, the Secretary of Agriculture, the Secretary of State; for education is the vital fluid that courses through and sustains the life of the State. In the subjects to be taught, and the methods of teaching them, the teacher has his grandest field of usefulness. Without his skilled hand, little can be done, particularly in the lattei direction; and the subjects of mathematics are, probably, receiv- ing less, and are in need of more, attention to-day than any other subjects in our public school curriculum. The gradual but per- manent changes that have been wrought in mercantile methods call for corresponding changes in instruction in arithmetic. Many of the subjects which are treated of in nearly all text- books upon arithmetic, might, with great advantage, be given much less time and space, or omitted altogether. This, to the so- called conservative, will sound radical and almost revolutionary. But a moment only of serious and candid thought will be required to convince such that it is in strict accordance with the intent and purpose of our public schools, ' ' The greatest good to the greatest number." Elementary geometry and concurrent industrial drawing could be taught with much practical and refining influence much earlier than they are at present. These changes would be in line with the best thought of the world, and would do much to satisfy the increasing demand for manual training in our public schools. Dr. Harris says, "Industrial drawing should have its place in the common school side by side with penmanship." He further says, "Culture and taste, such as drawing gives, fits all laborers for more lucrative situations and helps our produc- tions to hold the markets of the world." Professor Warren, after pointing out in detail and at great length the uses and purposes of geometry, concludes by saying, "From all these considerations, we may conclude, without rashness, that to not RESUME. 161 less than half of the 37,000,000 of industrial age, more or less knowledge of geometry, as early and as simply begun as arithmetic commonly is, would be highly beneficial." This number embraces more that half the entire population of the United States; hence, the conclusion of Prof. Warren, if it be true, is a strong one; and as it was made in the Forum, and stands uncontradicted, it may safely be quoted as not far from the truth. In too many schools, however the pupil may be taught at first, as he advances, his arithmetic exercises deteriorate to mere mechanical work on slate, paper, or blackboard; never a clear cut explanation from the pupil, or a " why ' ' from the teacher. The pupil is never asked or encouraged to solve his problem in another way, is never encouraged to find a shorter method and, therefore, a better one; hence, one of the most effectual ways of arousing enthusiasm, of begetting confidence, of inducing expression of thought, is not called into requisition. The motto in arithmetic should be, "the shortest solutions and the best explanations." With this end in view, a variety of solu- tions should be encouraged,' that the pupil may be lead to com- pare them and select the best. In this way a love for the study and an enthusiasm will be developed which will certainly lead up to a clear conception of the subject. The very act of search- ing for short solutions constitutes one of the greatest stimuli to mental activity that can be imagined. It gives play to genius; it undoes the levels of the graded schools; it harms none; it benefits all, some inestimably. Explanations should always be required for the sake of clearness of comprehension of the problem itself, for the sake of the development of the powers of close and exact expression, and for the sake of building up confidence. Pupils need to stand upon their feet and to talk in every study, so that their language may be criticized and clarified and purified. The short solutions will result in so much time saved that the number of explanations may be increased and their character greatly improved without employ- ing any additional time. There is also room for great improvement in setting forth the objects of the study. Boys, particularly, and all to a greater or less degree, are ever wanting to know of what use a particular study will be; and, certainly, they ought to be told, for it is 162 RESUME. always possible to do so. These ' ' whys ' ' are the germs of knowledge that should be watered with the dews of reason and patience, that from them may grow the oaks of wisdom and mental power. Boys love the practical; they early feel that a great responsibility rests upon their shoulders; and they should be encouraged so to feel, and assisted to prepare for their great life work. They should be convinced, as they all can be, that they will not only be wiser and better, but that their chances for making money will be improved by a thorough education. The teacher is never without opportunities for improvement. There is, among the many others, a great work before him the faithful performance of which will redound more to the monetary and industrial advancement of the country than it is within the capabilities of man to estimate. This work can be done thor- oughly and well by the teacher, and by him only. It is a work concerning which there is a consensus of educated opinion; and yet it seems no nearer accomplishment to-day than twenty -five years ago. Not so near, in fact; for it was then that Congress authorized the use of the Metric System of Weights and Measures in the United States. It was then ,that under the stimulus of this Act, its friends were led to hope, and believe even, that its introduction and use would be easily and speedily accomplished. The real significance of the law was over-estimated. It was, in fact, no law at all. It contained no compulsory provisions. It contained nothing that was tangible. It was like a plank in a party platform, pleasing to the ear, but possessing no legal, or even moral, force within its provisions. The friends of the Metric System were lulled to silence and inactivity by it, and the cause of education in this country was retarded in that direction many years, how many no one can tell. The Metric System is of such incalculable value that no effort should be spared that will hasten its final and universal use. It should, therefore, be taught in every school in the land; it should be compared with the cumbersome and illogical old system in the presence of every man, woman, and child in the nation; its praises should be sung in the ears of our people whenever and wherever oppor- tunity offers; its advantages should be mathematically demon- strated and estimated in the years, months, and days of a person's life-time, and in the dollars and cents that can provide so many comforts and luxuries. The Metric System can be thoroughly RESUME. 163 taught in one week. Of course it cannot be done in so short time, if it is required to change from the units of one system to those of the other; but why should that be taught at all? The very act of thus teaching it is what has retarded its introduction. That very mistaken idea of teaching it has made it appear to be beset with difficulties, when in fact it is analogous to our money system, the simplest in the world. Who would think of return- ing to our old system of shillings and pence ? How ridiculous to suggest such a thought even ! Yet, no more ridiculous than to resist, or rather not to encourage, the adoption of its comple- ment and coadjutor, the Metric System of Weights and Measures. It is the system used by almost the entire civilized world. It is the system the necessity for which John Quincy Adams saw as early as 1821, when he said; "Uniformity of weights and meas- ures, permanent, universal uniformity adapted to the nature of things, to the physical organization and the moral improvement of man, would be a blessing of such transcendent magnitude, that if there existed upon earth a combination of power and will adequate to accomplish the result by the energy of a single act, the being who should exercise it would be among the greatest of benefactors to the human race. ' ' The subjects of Square and Cube Root, Arithmetical and Geometrical Progressions, and Mensuration will receive attention in ' ' Pedagogics Applied to Algebra ' ' and in ' ' Pedagogics Applied to Geometry" now in the course of preparation. THE END. I A 000937153 5