LESSONS ON NUMBER. THE MASTER'S MAN! >s. Gd. IN MEMORIAM FLOR1AN CAJOR1 LESSONS ON NUMBER. ( LESSONS ON NUMBER, IN A PESTALOZZIAN SCHOOL, CHEAM, SURREY. THE MASTER'S MANUAL. BY C. REINER, TEACHER OF MATHEMATICS IN CHEAM SCHOOL. LONDON: PRINTED FOR JOHN TAYLOR, BOOKSELLER AND PUBLISHER TO THE UNIVERSITY OF LONDON, UPPER GOWER STREET. 1835. ^ PREFACE. NUMBER presents a most important field, on which to develope and strengthen the minds of children. Its obvious connec- tion with the circumstances surrounding them, the simplicity of its data, the clearness and certainty of its processes, -the neatness and indisputable correct- ness of its results, adapt it in an emi- nent degree for early instruction. Arith- metical exercises tend to give clearness, activity, and tenacity to the mind ; many an intellect that has not power enough for geometry, nor refinement enough for language, finds in them a department of study, on which it may labour with the invigorating consciousness of success. A 3 Vi PREFACE. But the advantages must of course de- pend, in a great measure, on the manner in which arithmetic is taught. More than any other branch of instruction has it suffered, in this country, from the in- fluence of circumstances. The reproach, that we are a nation of shopkeepers, might seem to have originated in the spirit of our arithmetical studies. Most popular treatises on the subject degrade the science they profess to elucidate ; it is made a mere shop-boy's assistant, the A certain mechanical dexterity in per- forming the operations of arithmetic, as required by the circumstances of com- mercial life, is effectually obtained by the use of these treatises; but the prin-^ ciples of , the science jare_ unknown, and many of its advantages, as presenting an exercise of mental power, altogether neglected. PREFACE. Vll Intelligent teachers, however, have not always been satisfied that their pupils should regard their mode of calculating as correct, or convenient, merely be- cause it corresponded with the rule given in their book ; they have explained the rationale of the process. The simple^ lucid, and well-arranged treatise of Pro- fessor I)e Morgan, is among the happiest attempts to rescue arithmetic from its present degraded state, and to claim for it a place among other branches of ra- tional education. It is peculiarly valu- able for young persons, who, having been from their infancy led hood-winked ]\ ( through the dark alleys of arithmetical ] rules, desire to take an intellectual view_ of operations, which they have been taught to perform mechanica,lly. It takes them, as it were, to an eminence, where^\ they can see the point from which they started, and that at which they have viii PREFACE. arrived, and, tracing all the windings of the dark passages which they were made to traverse, shows them that they were indeed the shortest, if not the best course they could have followed. The aim of the little work now offered to the public, is different ; it does not pro- pose to explain processes, but to unfold principles. The pupil is not taught to ^comprehend a rule, butjo L dispense : with it, or form it for himself^ The path along which he is led may be longer than the usual route, but then it is in broad day- light ; he is more independent of his guide, and derives more health and vi- gour from the exercise. Were the true ends of intellectual education more clear- ly apprehended, the means of prosecuting it would be more justly appreciated. While the question cui bono ? so judicious in itself, is answered by a sordid refer- ence to mere money-getting, or by a PREFACE. IX narrow-minded consideration of profes- ^ional advancement, every method of in- struction that proposes to itself a more exalted, though less obvious utility, will be ridiculed as visionary, or neglected as unprofitable. But when the true end of intellectual education shall be admitted to be, first, the attainment of mental power, and, then, the application of it to\\ practical and scientific purposes, that] plan of early instruction, which dwells long on first principles, and does not haste to make learned, will be acknow- ledged as the most economical, because the most effectual. Experience will show, as indeed it has already shown, that while superficial teaching may pre- pare for the mere routine of daily busi- ness, whensoever a question, not anti- cipated in the manual, occurs, none but the pupil whose faculties have been exer- cised in the investigation of truth, who is X PREFACE. the master, not the slave of rules, will solve the unexpected difficulty, by a novel application of the principles of the science. Writers on method have observed, that there is a certain order, in which truths present themselves to the mind engaged in the original investigation of a subject, and that when the subject has been investigated, a different arrange- ment is necessary for the lucid exposi- tion of the truths discovered. These views have been most unhappily applied in the early stages of instruction. For although the artificial order may be best calculated to convey knowledge to minds already trained for its reception, by pre- vious acquaintance with similar subjects, it is by no means suited to the opening faculties of children. Hence the dis- gust, in many cases insurmountable, which the first principles of a science in- PREFACE. X spire in their minds. This disgust, how- ever, vanishes, if a preparatory course of instruction be arranged, having for its object the training the mind for the study of the science rather than the com- municating the knowledge of it. In this preparatory course the order is deter- mined by a consideration of the mind of the pupil ; it commences with what is already known to him, and proceeds to the proximate truth ; the more easy pre- cedes the more difficult, the individual prepares for the general truth, the ex- ample for the rule. It has been objected to the former edition of this little work that it is not complete ; it does not profess to be so ; it is only a preparatory course; the vesti- bule, not the building. It is proposed that rules, and practice on rules, should follow these exercises ; and this plan has for years been adopted in the school, for Xll PREFACE. the use of which the lessons were ori- ginally drawn up. The principal alter- ation which the Second Edition exhibits, is the division of the work into two Parts, the one supplying Directions for the Master, and the other Exercises for the Pupil. This arrangement will mate- rially diminish the manual labour of the teacher, and facilitate his giving instruc- tion to several classes at the same time. It is obvious that a single copy of the larger part will suffice for the Master, and that each of the pupils should be furnished with a copy of the Praxis. C. MAYO. Cheam, Jan. 1, 1835. LESSONS ON NUMBER. INTRODUCTION. ON NUMBER. Idea of Number Counting. THE aim of this lesson is, to lead the pupils to understand the meaning of words, expressive of numbers, as " two/' " three," " ten/' &c. ; namely, that they imply a collection of so many ones. To do this, any convenient object, such as a ball, stone, book, &c. being placed before the pupils, the Teacher says One ball. Pupils. [Repeat.] One ball. T. Show me any other object in the room of which you may say one. P. One ceiling, floor, 8cc. T. (Putting a second ball to the former, says) Two balls. P. [Repeat.] Two balls. 2 LESSONS ON NUMBER. T. Show me any objects of which you may say two. P. Two doors, &c. 'T. (Adding a third ball to the former two, says) Three balls. P. Three balls. T. Name any objects of which you may say three. P. Three chairs, &c. Adding successively a ball to the former number, the teacher each time requires the pupils to point out some objects to which the appellation, four, five, six, &c. is applicable. It must be left to his dis- cretion to determine where to stop. One child will be embarrassed by having ten or twenty ob- jects before him, whereas another will, at one glance, ascertain their number. In the one the power of perception must be developed by a slow and gradual process ; in the other, it will rapidly strengthen as larger collections of objects are suc- cessively presented. In order however to lead to the accurate con- ception and correct expression of number, it is desirable at this step to put before the pupils pro- miscuous objects, requiring them to ascertain their total number, as well as the number of those among them, which are of the same kind. Thus, for instance ; putting 3 balls, 4 books, 5 slates, &c. before them, the teacher says LESSONS ON NUMBER. 3 T. What must you do to ascertain how many objects there are here? P. We must count them. T. Count them. P. There are twelve objects. T. Twelve, then, is the number of objects before you ; are they all of the same kind ? P. No ; there are balls, books, and slates. T. Ascertain the number of each class of objects. P. The number of balls is three ; the number of books four ; and the number of slates five. Exercises of this kind may be much and inte- restingly diversified, especially for very young children, or for those whose perception is slow, by directing their attention to the number of the various parts of which one object is composed, or the number of plants, trees, birds, fishes, shells, minerals, &c. which they know. The teacher is referred to " Exercises on Lessons on Number." B 2 CHAPTER I. - ADDITION. 1. ADDITION OF UNITS. LESSON I. To add One. THE first and simplest lesson in Number is, evi- dently, that in which the pupil is taught to add one to each number in succession ; and then to numbers taken promiscuously. To connect this operation of the mind with the idea the pupil has formed of number from the previous exercises, recourse must be had to the senses, by making use of some visible objects, either the numeral frame or the slate. This being ultimately aban- doned, the exercise will become purely intellectual. In the following exercises the slate is supposed to be used. T. [Drawing one small line upon the slate, asks] How many lines are there here ? P. One. T. [Draws beneath the former two lines, and asks] How many lines are there here? | | P. Two. ADDITION. 5 T. How many are one line more one line ? P. Two lines. T. How many are one book more one book ? One tree more one tree ? One slate more one slate? How much are one more one ? How many ones make two ? T. [Drawing three lines beneath the former two, asks] How many lines are here ? Ill P. Three. T. How many more lines are there in this row than in the one above ? P. One more. T. How many lines are in the second row ? P. Two. T. How many lines are two more one ? P. Three lines. T. How much is two more one ? P. Three. T. [Draws four lines beneath the former three, asking] How many more lines are there in this row than in the one above ? I I I I P. One more. T. How many are in the third row ? P. Three. T. How many are three lines more one ? P. Four lines. T. How much is three more one ? P. Four. T. [Draws Jive lines beneath the former four ; szV, seven lines, and so on in succession, asking 6 LESSONS ON NUMBER, each time questions analogous to the above. These lines will stand arranged thus : To such a number, as, in the discretion of the teacher, will seem sufficient. After due repetition, the pupils must be able to proceed as follows : [the lines on the slate being effaced.] One more one are two ; Two more one are three ; Three more one are four ; Four more one are five ; Five more one are six ; Six more one are seven ; &c. &c. &c. The teacher will do well to let his pupils proceed to the very limit of their distinct conception of number. After this exercise, which the whole class should repeat viva voce, each pupil may be ADDITION. / required to ask the class one or more questions promiscuously. Thus, How much is nine more one ? Seventeen more one ? &c. It frequently happens that the teacher's class consists of several divisions of pupils, more or less advanced, and the time he can devote to each is, in consequence, limited. Exercises have, there- fore, been drawn lip corresponding to this and the following lessons, which may be given to the pupils for solution. The answers are, for the pre- sent, to be written on their slates in words. Another advantage of these exercises, and that not perhaps the least, is, that they lead each pupil in silence to think for himself, and commit the result of his thoughts to writing ; a process much calculated to sober the mind after the exciting effects of simultaneous learning. Answers to the Exercises. Lesson I. To add One. N. B. The answers to the six first questions must be inspected by the teacher. 1. 15 Ans. 13. 74 Ans. 19. 101 8. 27 14. 79 20. 109 9. 38 15. 81 21. 127 10. 49 16. 87 22. 149 11. 56 17. 94 23. 155 12. 63 18. 97 24. 187 LESSONS ON NUMBER. LESSON II. To add Two. T. [Drawing one line on the slate, and next to it two lines more, thus] asks How many lines are there in all ? P. Three lines. T. [Drawing two lines, and next to them two lines more, three lines, four lines, &,c. adding each time two lines, as below,] asks Mill MINI How many lines are one more two ? two more two? three more two lines? &c. &c. P. Two lines more two lines are four lines ; Three lines more two lines are five lines ; Four lines more two lines are six lines ; &c. &c. After due repetition of the above, the drawing of lines on the slate should be abandoned, if possible. It is far more improving to render the operation purely intellectual, than to continue deriving assist- ance from the external senses. ADDITION. To add two, is, evidently, to add one, and then another one in succession. If, then, Lesson I. be perfectly known, there will be little difficulty in leading the pupil to increase any number by two at one step. The mode made use of to this end, is shown in the following questions : Q. How many ones make two ? A. One more one ; that is, two ones. Q. How many ones, then, make one more two ? A. One more one, more one ; that is, three. Q. How much is two more two? A. Two more one, more one, or four. Q. How much is three more two ? A. Three more one, more one, or five. Q. How much is four more two ? five more two ? six more two? &c. The pupils must, after due repetition of such exercises, be able to proceed readily as follows : One more two are three; Two more two are four ; Three more two are five ; Four more two are six ; Five more two are seven ; &c. &c. For exercise, it is recommended to require the pupils to count by twos, as they have learned to count by ones ; that is, the teacher begins at any number he pleases, and calls upon the pupils to add two, and to the answer two again, and so on in succession, as far as they are able, or as may be sufficient for practice. B 5 10 LESSONS ON NUMBER. Answers to the Exercises. Lesson II. Ans. 1. 19 Am. 5. 63 Am. 9. 103 2. 27 6. 71 10. 121 3. 49 7, 82 11. 151 4. 58 8. 91 12. 201 Am. 13. 1, 3, 5, 7, 9, 1 1, 13, 15, 17, 19, 21, 23, &c. Am. 14. 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, &c. LESSON III. To add Three. Before entering upon any new lesson, the teacher must be quite certain that the preceding is firmly fixed in the mind of the pupils. On this supposi- tion, the present lesson will require but little effort on their part. They are able to add two, also to add one, and are now required to add three; that is, two more one. Teacher. You have learnt to add two; if, now, instead of two, you were required to add three, what would you do ? Pupils. First add two, and then one more. T. How much is one more three ? P. One more two, more one, or four. T. How much is two more three ? P. Two more two, more one, or five. T. How much is three more three? ADDITION. 11 P. Three more two, more one, or six. T. How much is four more three? five more three? six more three? &c. &c. After due repetition, the pupil must be able to proceed readily thus : One more three are four ; Two more three are five ; Three more three are six ; Four more three are seven ; &c. &c. The teacher now may call upon the pupils to give questions promiscuously to the class. Pupil How much is 1 7 more 3 ? Class. Twenty. P. How much is 59 more 3 ? C. Sixty-two. &c. &c. The whole class then begin to count by adding three in succession ; thus: C. Three, six, nine, twelve, fifteen, 'Sec. Again, C. One, four, seven, ten, thirteen, &c. And again, C. Two, five, eight, eleven, fourteen, &,c. As a repetition and preparation for the next lesson, the teacher may combine the previous two lessons with the last. Thus : T. How much is one more three, more one? two more three, more one ? four more three, more one ? five more three, more one ? 8cc. &c. LESSONS ON NUMBER. Again, r. How much is one more two, more two? two more two, more two ? three more two, more two ? &c. &c. And again, T. How much is one more two, more three? two more two, more three ? three more three, more three ? &c. Then, promiscuously : T. How much is five more three, more two? seventeen more two, more two ? thirty-nine more two, more three ? forty -five more one, more two, more three ? It will much tend to enliven the pupils, if the teacher permit them to give questions, similar to the above, to the class. For instance, one pupil asks, How much is forty -three more three, more two ? Those who have found the answer, hold up one hand. The pupil who gave the question chooses among them whom he pleases to answer it; and it is he, likewise, who is to say whether the answer be right or wrong. ADDITION. 13 Answers to the Exercises. Lesson III. Am. I. 17 Ans. 4. 91 Ans.l. 60 2. 32 5. 78 8. 102 3. 40 6. 101 9. 121 10. 152 Am. 11. 4,7, 10, 13,16, 19, &c. 12. 5,8, 11, 14, 17, 20, &c. 13. 6, 9, 12, 15, 18, 21, &c. Ans.U. 10 Ans. 16. 41 Ans. 18. 92 15. 23 17. 62 19. 104 20. 96 LESSON IV. To add Four. This and the following lessons are quite analo- gous to those preceding ; and it is not without reason that the teacher is recommended the ob- serving a progressive order in these lessons, always beginning with the simplest form possible. Accord- ingly, the number four is added to one, then to two, next to three, and so on in succession, as far as the teacher may think proper, within the limit of the pupil's clear conception of number. The answers, being the numbers 5, 6, 7, &c. in their natural order, are readily found by the pupils, which serves as encouragement; but to arrive at the aim in view, namely, to add four to any num- ber at one step, the teacher requires the pupils to 14 LESSONS ON NUMBER. count by fours; that is, he begins with adding four to one; to the answer four again is added, and so on, which gives rise to the following pro- gression : 1, 5, 9, 13, 17, 21, 25, 29, 31, &c. Two is then taken as the beginning of the next series ; thence the following progression : 2, 6, 10, 14, 18, 22, 26, 30, 34, &c. Three now begins the series, whence the an- swers, 3, 7, 11, 15, 19, 23, 27, 31, 35, &c. And, finally, four is made the beginning from which the answers 4, 8, 12, 16, 20, 24, 28, 32, 36, &c. It is obvious, that by this mode of proceeding, four is added to each of the numbers, 1, 2, 3, 4, &c. and the subject quite exhausted. The pupils must be able to repeat these four series, with- out hesitating, before they proceed to the next lesson. Promiscuous questions for practice are left to be given by each pupil to the class. Answers to the Exercises. Lesson IV. Ans. 1. 43, 82, 91, 99, 123. 2. 22 Ans. 4. 66 Ans. 6. 107 3. 43 5. 95 7. 43, 47, 51, 55, 59, 63, &c. 8. 61, 64, 66, 70, 73, 75, 79, 82, 84, 88, 91, 93, 97, 100, 102, 106, &c. ADDITION. 15 LESSON V. To add Five. The difficulty of adding a number at one step increases with the number itself ; but it may be removed in a great measure, by inducing the pupils to use the knowledge already gained. Thus to add five is evidently to add four more one ; or to add three more two ; or two more two more one ; ope- rations already performed by them. The teacher ought therefore to object to the pupils counting by their fingers, but constantly require them to refer to that actually known, and to use it accordingly. The teacher may proceed thus : - Teacher. You have learnt to add four ; if now it be required to add five, what would you do ? Pupil. First add four, and then one more. T. But if you began with adding three, what then must be done ? P. Add two more. T. How much then is seventeen more five ? P. Seventeen more four more one, that is twenty-two. T. Or by adding three first. P. Seventeen more three more two, which is twenty-two. T. The answer twenty-two is usually called the sum of seventeen more five ; and whenever two or more numbers are added together, the answer is 16 LESSONS ON NUMBER. called their sum. What is the sum of thirty-seven more five? P. Forty-two. T. What is the sum of one, two, three, four, and five? P. Fifteen. We will now learn to add five at one step, without first adding four and then one, or three and then two. Here now follow exercises analogous to those in the preceding lessons. T. How much is one more five ? two more five ? three more five ? four more five ? 8cc. &c. After which one is made the beginning of the series to which five is added, to the sum five again, arid so on. Then two is made the beginning, three next, four next, and finally five, which complete the lesson. The answers, which the pupils must be able to repeat viva voce, are respectively as follow : Class, count 1, 6, 11, 16, 21, 26, 31, 36, 41, &c. 2, 7, 12, 17, 22, 27, 32, 37, 42, &c. 3, 8, 13, 18, 23, 28, 33, 38, 43, &c. 4, 9, 14, 19, 24, 29, 34, 39, 44, &c. 5, 10, 15, 20, 25, 30, 35, 40, 45, &c. This mode of proceeding will soon be remarked by the pupil ; he will anticipate the questions of ADDITION. 17 his master ; it will awaken his mind to reflection, and lead him to self-activity. Before entering upon the next lesson, each pupil gives questions to the class, making now use of the word sum. Thus : P. What is the sum of fifty-eight and five ? &c. Answers to the Exercises. Lesson V. Ans. 1. 48 Am. 4. 112 Ans. 7. 1.14 2. 69 5. 151 8. 203 3. 99 6. 162 9. 224 10. 22, 26, 29, 31, 32, 37, 41, 44, 46, 47, 52, 56, 59, 61, 62, 67, 71, 74, 76, 77, See. LESSON VI. To add Six. The mode of proceeding being quite in accord- ance with that detailed in the foregoing lessons, it will be sufficient to give a general outline of those which follow in this paragraph. 1st, Teacher. How much is one more six? two more six ? three more six ? &c. &c. 2d, Class, count 1, 7, 13, 19, 25, 31, &c. . . 2, 8, 14, 20, 26, 32, &c. .. 3, 9, 15, 21, 27, 33, &c. .. 4,10,16,22, 28, 34, &c. .. 5,11,17,23, 29, 35, &c. Finally, . . 6, 12, 18, 24, 30, 36, &c. 18 LESSONS ON NUMBER. 3d. Promiscuous questions given by each pupil to the Class. Expl. 1. Pupil. How much is fifty-nine more six? Expl. 2. P. Add seventy-three and six. Expl. 3. P. What is the sum of thirty-seven and six ? Expl. 4. P. What is the sum of eight, six, five, and four ? N. B. The pupils should be required to use the words, more," " add," " sum," in order to form correct notions of their signification. Answers to the Exercises. Lesson VI. Am. 1. 53 Ans. 6. 76 Ans. 11. 89 2. 21 7. 126 12. 93 3. 110 8. 56 13. 110 4. 118 9. 64 14. 116 5. 147 10. 78 15. 255 LESSON VII. To add Seven. Outline of the various Exercises of this Lesson. 1st. Teacher. How much is one more seven? . . two more seven ? . . three more seven ? &c. &c. ADDITION. 19 2d, Class, count 1, 8, 15, 22, 29, 36, 43, 50, &c. . . 2, 9, 16, 23, 30, 37, 44, 51, &c. . . 3,10,17, 24, 31, 38, 45, 52, &c. . . 4, 1 1, 18, 25, 32, 39, 46, 53, &c. . . 5, 12, 1 9, 26, 33, 40, 47, 54, &c. .. 6,13,20,27, 34,41, 48, 55, &c. Finally, . . 7,14,21, 28, 35, 42, 49, 56, 8cc. 3d. Promiscuous questions given by the pupils : Expl. 1. Pupil. How much is forty-three more seven ? Expl. 2. P. Increase seventy-seven by seven. Expl. 3. P. Add eighty-nine and seven. Expl. 4. P. Find the sum of ninety-eight and seven. Expl. 5. P. Find the sum of one, two, three, four, five, six, and seven. Answers to the Exercises. Lesson VII. Am. 1. 67 Am. 6. 83 Am. 11. 66 2. 77 7. 90 12. 132 3. Ill 8. 114 13. 127 4. 124 9. 122 14. 201 5. 129 10. 201 15. 326 LESSON VIII. To add Eight. Outline of this Lesson. 1st. How much is one more eight? two more eight ? . , , , three more eight ? 20 LESSONS ON NUMBER. 2d, Class, count 1, 9, 17, 25, 33, 41, &c. 2, 10, 18, 26, 34, 42, &c. 3, 11,19, 27,35, 43, &c. 4, 12. 20, 28, 36, 44, &c. 5, 13, 21, 29, 37, 45, &c. 6, 14, 22, 30, 38, 46, &c. 7, 15, 23, 31, 39, 47, &c. Finally, . . 8, 16, 24, 32, 40, 48, &c. 3d. Promiscuous questions given by the pupils : Expl. 1. How much is forty-five more eight? Expl. 2. Increase seventy-eight by eight. Expl. 3. Add eighty-seven and eight. Expl. 4. Find the sum of ninety-six and eight. Expl. 5. Find the sum of two, seven, five, and eight. Answers to the Exercises. Lesson VIII. Am. 1. 83 Ans. 6. 131 Ans. 11. 65 2. 109 7. 128 12. 83 3. 124 8. 131 13. 89 4. 143 9. 204 14. 122 5. 251 10. 335 15. 134 LESSON IX. To add Nine. Outline of this Lesson. 1st, Teacher. How much is one more nine? two more nine ? three more nine ? &c. &c. ADDITION. 21 2d, Class, count 1, 10, 19, 28, 37, &c. .. 2, 11, 20, 29, 38, &c. .. 3, 12, 21, 30, 39, &c. .. 4, 13, 22, 31, 40, &c. . . 5, 14, 23, 32, 41, &c. . . 6, 15, 24, 33, 42, 8cc. .. 7, 16, 25, 34, 43, &c. .. 8, 17, 26, 35, 44, &c. Finally, . . 9, 18, 27, 36, 45, &c. 3d. Promiscuous questions given by the pupils : Expl. 1. Pupil. How much is sixty-three more nine ? Expl. 2. P. Increase forty-seven by nine. Expl. 3. P. Add seventy-eight and nine. Expl. 4. P. What is the sum of ninety-two and nine? Expl. 5. P. Find the sum of thirteen, nine, seven, and six. Answers to the Exercises. Lesson IX. Am. 1. 50 Ans.6. 72 Am. II. 193 2. 100 7. 84 12. 224 3. 99 8. 129 13. 243 4. 116 9. 101 14. 262 5. 174 10. 138 15. 431 LESSON X. Decomposition of Numbers into Tens mid Units. The decimal system of numeration being that generally adopted, it is of importance to lead the 22 LESSONS ON NUMBER. pupils, as far as their tender years allow, to the perception of its advantages : and this may be done at the present stage. Teacher. Count as far as ten. Pupils. [Counting.] One, two, ten. T. The number next above ten is eleven, that is, ten more ? P. Ten more one. T. The number following is twelve ; that is P. Ten more two. T. Continue counting thirteen is P. Thirteen is ten more three ; Fourteen is ten more four ; Fifteen is ten more five ; Sixteen is ten more six ; Seventeen is ten more seven ; Eighteen is ten more eight; Nineteen is ten more nine ; Twenty is ten more ten ; Twenty-one is ten more eleven. jf. You said before, that eleven was ten more one ; what, then, will you say that twenty- one is ? P. Ten more ten, more one. T. That is, how many tens ? P. Two tens more one. T. Continue, twenty-two is P. Twenty-two is two tens more two. T. Two what? P. Two ones. ADDITION. 23 T. Proceed : twenty-three is P. Twenty-three is two tens more three ones ; Twenty-four is two tens more four ones, &c. &c. Twenty-nine is two tens more nine ones. T. Instead of saying ones, it is usual to say units*. Proceed. P. Thirty is two tens more ten units. T. That is? P. Three tens. Thirty-one is three tens more one unit, 8cc. &c. Forty is three tens more ten units ; that is, four tens. The pupils are required to continue, in a similar manner, as far as the teacher may judge neces- sary. T. Hence it is easy to say of how many tens and units a number consists. Thus, eighty-eight consists of how many units? P. Of eighty-eight units. T. And of how many tens ? P. Of eight tens and eight units. T. By which part of your body are you apt to count? P. By our fingers. T. And how many fingers have we? P. Ten. T. Whence, then, do you think originates our counting thus by tens ? 24 LESSONS ON NUMBER. P. Probably from the circumstance of our hav- ing ten fingers. T. How many tens are in eleven more ten? P. Two tens more one unit. T. How many units is that ? P. Twenty-one. T. Continue adding ten to twelve, thirteen, &c- P. Twelve more ten is two tens more two units, or twenty-two ; Thirteen more ten is two tens more three, or twenty-three ; Fourteen more ten is two tens more four, or twenty-four. Fifteen, &c. T. Hence, if ten is to be added to a number, what must be done ? P. Only increase the number of tens by one ten. T. Give an example. P. Thirty-eight more ten, is three tens more eight units, more one ten ; that is, four tens more eight units, or forty-eight. It will be convenient, before proceeding further, to teach the pupils how to represent numbers by signs, not that they are essential to the succeeding lessons, which are intended as practice in mental arithmetic ; but that the written exercises would, without the aid of signs, become cumbrous. In a school, then, where the teacher's time is divided between several classes, more or less advanced, the ADDITION. 25 necessity of the exercises will be felt ; and for such, it is recommended to have recourse to written arithmetic : whereas, in a family of a few children, no such necessity existing, it will be better to defer it, and continue the exercises purely mental, until the pupils' minds, from the following lessons, are more developed. How to represent Units by Signs. I. UNITS. Teacher. We will now learn to represent num- bers by signs [writing the following on the large school-slate] : 2. 3. 4. 5. 6. 7. 8. M M M I [Pointing to the lines, and their respective repre- sentatives in number, says, and pupils repeat], One, two, three, &c. These signs are called Jigures ; imitate them on your own slates. c 26 LESSONS ON NUMBER. jT. What numbers do these figures represent? Pupils. Ones, or units. T. Hence the figure 2 is equal in value to which two figures added together ? P. To 1 more 1. T. I will teach you now another sign, generally used to denote the word "more;" it is this, + . If, then, we wish to write down what we have just said, we would, instead of writing thus, 2 is equal in value to 1 more 1, write, 2 is equal in value to 1 -f- 1. For the words, " is equal in value/' or " are equal in value," it has been agreed to substitute the sign = ; so that 2 is equal in value to 1 more 1, is written, 2 = 1 + 1- What advantage has this sort of short-hand writing ? P. It saves time and room. T. Let one of you come here to the slate, and, by means of these signs, write which three num- bers the figure 3 is equal to. P. [Writing.] 3 = 1 + 1 + 1. T. And which two figures is 3 equal to ? P. [Writing.] 3=2 + 1. T. Proceed in a similar manner-with the figure 4. P. [Writing.] 4 = 1 + 1 + 1 + 1. 4=2 + 1 + 1. 4=2 + 2. 4=3 + 1. ADDITION. 27 T. Let, now, every one of you proceed in the same way with the remaining figures, 5, 6, 7, 8, and 9, on your own slates. II. TKNS. T. What is the number next above nine? P. Ten. T. Represent ten by the addition of two of the figures you have learnt. P. 9 + 1; 8 + 2; 7 + 3, &c. T. How many ones, or units, are there in ten ? P. Ten units. T. And how many tens? P. One ten. T. Which of the figures does represent one? P. 1. T. If, now, we wish to represent one ten, will it be sufficient to write 1 ? P. No ; because we could not tell whether one was meant or ten. T. In order to raise this difficulty, I will place this sign, 0, called zero, next to the 1, showing that the 1 is meant for 1 ten; thus, 10. Tell me, now, which place occupies with respect to the 1? P. It stands behind the 1 ; it stands before the 1. T. Some of you said, " behind,'' others, " be- fore;" which is right? c 2 28 LESSONS ON NUMBER. P. " Behind ;" because we read and write from the left to the right. T. So we do with the letters of the alphabet ; but are you sure it is the same when signs are used instead of letters? You mentioned "left" and " right;" which of these places does occupy in respect to 1 ? P. It stands to the right of it. T. Well ; tell me, now, how many units there are in 10 ? P. Ten units. T. And how many tens? P. One ten. T. So that if we suppose these ten units col- lected, there would be one ten, and no more units. Which, now, of the figures 10, indicates one ten, and which no units ? P. The one in 10 indicates one ten, and the zero no units. T. And you have before remarked that the figure zero stands where? P. To the right of 1. T. Hence we shall for the future agree, that, of two figures, that which stands to the right of the other, indicates what? P. Units. T. And what to the left? P. Ten. T. How am I to represent two tens, or twenty ? ADDITION. 29 P. Write the figure two, and to the right of it zero. T. Let one of you come here, and write three tens, or thirty ; four tens, or forty ; fifty, sixty, &c. ninety. T. How many tens are there in one hun- dred ? P. Ten tens. T. You have learnt to write ten ; we have agreed upon the place which the tens should occupy- try and express one hundred. It must be left to the pupils to represent that number correctly; when done, the teacher pro- ceeds : T. How many figures have you used to repre- sent one hundred ? P. Three figures. T. How many units are there in one hun- dred ? P. One hundred units. T. How many tens ? P. Ten tens. T. And how many hundreds? P. One hundred. T. So that if we suppose these hundred units collected into tens, there would be how many of them? P. Ten, and no units. 30 LESSONS ON NUMBER. T. And again ; if we suppose these ten tens col- lected into hundreds, there would be P. One of them, and no tens. T. Which, now, of the figures 100, indicates no units, which no tens, and which one hun- dred ? P. The last zero to the right indicates no units \ the next, to the left of the former, indicates no tens ; and the one to the left of this indicates one hundred. T. Hence we will for the future agree, that, of three figures, that which stands to the right of the two others, shall indicate P. Units. T. And that which stands next to the left of the former, shall indicate P. Tens. T. Finally, that which stands to the left of this, shall indicate P. Hundreds. T. Now represent on your slates two hundred, three hundred, four hundred, &c. From this it will be readily perceived how the pupils may be taught to represent thousands, tens of thousands, &c. ; the teacher, remembering that the mind of the pupil will be strengthened by pur- suing these exercises, so long as he conceives with clearness, but no longer. ADDITION. 31 As before with units, so now the pupils are re- quired to proceed with tens. Thus : 20=10+10. 30=10 + 10 + 10 =20 + 10. 40= 10 + 10 + 10+ 10 =20 + 10+10. =20 + 20 =30 + 10. &c. &c. III. TENS AND UNITS. Teacher. Mention the numbers which are be- tween ten and twenty. Pupils. Eleven, twelve, &c nineteen. T. Bearing in mind what places it was agreed that units and tens should occupy, try to re- present the number eleven. The pupils must be left to represent that number correctly. The right or the wrong the class must decide ; for that purpose it is best to call each in turn to the school slate. Should eleven be written thus 101, the teacher asks T. We agreed that the last figure to the right shall represent units ; the next to the left of it tens ; the third hundreds, 8tc. Does this number represent eleven, &c. &c. until the correct answer 11 is found. T. Here then are two ones, the one to the right of the other indicates ? P. One unit. 32 LESSONS ON NUMBER. T. Ttjat figure is then said to stand in the unit place ; the other 1 to the left of the former in- dicates P. One ten. T. It is said to stand in the tens place; now re- present twelve, thirteen, &c. In a manner quite analogous to the above, the pupils are lead to represent the numbers, twenty- one, twenty-two, &c. &c. which will not be at- tended with any difficulty, if the preceding has been understood. The pupils must discover them- selves that all numbers between 1 and 100 are pro- perly represented by 2 figures, those between 100 and 1000 by 3 figures, and so on. For practice, it is recommended to require the pupils to write in figures the questions and their answers according to the following model : thus, beginning at Lesson I. Quest. 7. The pupils write on their 14+1 = 15. 26+1=27. 37 + 1=38. Lesson II. Quest. 1. 17 + 2 = 19. 25 + 2=27. 47 + 2=49. ADDITION. 33 Lesson III. Quest. 1. 14 + 3 = 17. 29-4-3=32. 37 + 3=40. &c. &c. And so on with the lessons which follow : Thus, Lesson IX. Quest. 1. 5 + 9 + 4 + 9 + 3 + 9 + 2 + 9=50. The mental exercises are now resumed ; but those which the pupils have to perform, as practice on their slates, are now written in figures instead of words. 2. ADDITION OF TENS. LESSON I. To add Ten. Teacher. How much is one more one ? Pupils. Two. T. How much is one ten more one ten ? P. Two tens. T. That is- P. Twenty units. T. How much is two more one ? P. Three. T. How much is two tens more one ten ? P. Three tens ; that is, thirty, c 5 34 LESSONS ON NUMBER. T. How much is three more one ? P. Four. T. How much is three tens more one ten? P. Four tens, or forty. The teacher thus continues, first, by asking how much is four more one, then four tens more one ten five more one, then five tens more one ten ; and so on, until the pupils are able to proceed readily. Thus: Pupils count : ten more ten are twenty ; twenty more ten are thirty; thirty more ten are forty ; &c. &c. On comparing this lesson with Lesson T. Addi- tion of Units, it will be found to correspond with it in every step ; with this difference, that there is an addition of tens instead of units. The lessons which follow in this paragraph, namely, those which refer to the addition of two tens, or twenty, three tens or thirty, &c. precisely follow the same course as those, which had for their object, to add two, three, &c. ; and treated in this manner, present but little or no difficulty. From this consideration, it is thought sufficient to give an outline of one or two lessons on this subject, and leave it to the teacher to supply those which remain. Answers to the Exercises. Lesson T. To add Ten. The teacher, on examining the slates, must find ADDITION. 35 the questions and their answers drawn up as fol- lows : Ans.l. 40 + 10=50. Am. 6. 150+10 = 160. 2. 70 + 10=80. 7. 160 + 10=170. 3. 80 + 10=90. . 8. 190 + 10=200. 4. 90 + 10=100. 9. 280+10=290. 5.120+10 = 130. 10. 360 + 10=370. LESSON II. To add Twenty. Teacher. How much is one more two ? Pupils. Three. T. How much is one ten more two tens ? P. Three tens, that is thirty. T. How much is two more two ? P. Four. T. How much is two tens more two tens ? P. Four tens, that is forty. The teacher continues questioning, in a similar manner, until the pupils are able to proceed readily. Thus:- Class count : ten more twenty are thirty; twenty more twenty are forty; thirty more twenty are fifty ; 8cc. &c. &c. Then follow promiscuous questions, given as usual by the pupils to the class. Thus : Pupil 1. How much is ninety more twenty ? 2. Find the sum of 270 more 20. 3. Add 380 and 20. " 36 LESSONS ON NUMBER. Answers to the Exercises. Lesson II. Model how the exercises are to be written by the pupils : Am. 1. 50-f 20 = 70 2. 90 + 20 + 10=120. Ans.3. 130 Ans.7. 180 4. 150 8. 210 5. 210 9. 280 6. 120 10. 390 LESSON III. To add Thirty. Teacher. How much is one more three? How much is one ten more three tens ? How much is ten more thirty ? The questions being answered satisfactorily, the teacher proceeds : T. How much is two more three ? How much is two tens more three tens? How much is twenty more thirty ? And so on with thirty more thirty, forty more thirty, &c. ; and it will be necessary to begin always with the addition of units, and then trans- ferring the same to the corresponding tens. The pupils must, at the end of this lesson, be able to proceed. Thus : Class count : ten more thirty are forty ; twenty more thirty are fifty ; thirty more thirty are sixty ; &c. &c. &c. . ADDITION. 37 Then follow promiscuous questions given by the pupil : Pupil 1. Add 90 and 30. 2. Find the sum of 50, 30, and 20. 3. Increase 180 by 30. Answers to the Exercises. Lesson III. Model Ans. 1 . 70 + 30 + 20 + = 130. Ans. 2. 200 Ans. 6. 180 Ans. 10. 500 3. 270 7. 340 11. 530 4. 300 8. 130 12. 650 5. 140 9. 370 LESSON IV. To add Forty. Answers to the Exercises. Lesson IV. Ans. 1. 100 Ans. 6. 310 2. 210 7. 390 3. 280 8. 590 4. 470 9. 670 5. 570 10. 760 LESSON V. To add Fifty. Answers to the Exercises. Lesson V. Ans.}. 150 Ans. 6. 250 2. 230 7. 450 3. 300 8. 600 4. 380 9. 710 5. 420 10. 1090 38 LESSONS ON NUMBER. LESSON VI. To add Sixty. Answers to the Exercises. Lesson VI. Am. I. 210 Ans. 6. 540 2. 250 7. 550 3. 350 8. 720 4. 330 9. 810 5. 410 10. 960 LESSON VII. To add Seventy. Answers to the Exercises. Lesson VII. Ans. 1. 280 Ans. 4. 390 2. 400 5. 640 ; 3. 390 6. 1090 LESSON VIII. To add Eighty. Answers to the Exercises. Lesson VIII. Ans. 1. 360 Ans. 4. 730 2. 490 5. 770 3. 580 6. 1190 LESSON IX. To add Ninety. Answers to the Exercises. Lesson IX. Ans. 1. 440 Ans. 4. 600 2. 500 5. 860 3. 460 6. 1210 ADDITION. 39 3. ADDITION OF TENS AND UNITS. LESSON I. To add 10, 11, 12 19. Thus prepared, the pupils may now begin to add numbers consisting of tens and units, commencing again with the simplest form ; thus : Teacher. How much is 1 more 10? .. 2 more 10? .. 3 more 10? &c. Sec. .. 10 more 10? .. 11 more 10? Pupils. 10 more 1, more 10 ; that is, 21. T. How much is 12 more 10? P. 10 more 2, more 10; that is, 22. T. How much is 13 more 10? P. 10 more 3, more 10 ; that is, 23. T. How much is 14 more 10? 15 more 10? &c. .. 20 more 10? .. 21 more 10? P. 20 more 1, more 10, or 31. I 7 . How much is 22 more 10 ? P. 20 more 2, more 10, or 32. T. How much is 23 more 10? .. 24 more 10? . . 25 more 10 ? 8cc. 31 more 10? 40 LESSONS ON NUMBER. P. 30 more 1, more 10, or 41. T. How much is 32 more 10 ? .. 33 more 10? 34 more 10 ? &c. &c. And so on with the numbers which follow. The pupils will soon be able to proceed readily thus : Class count: 44 more 10 are 54; 45 more 10 are 55 ; &c. 73 more 10 are 83 ; &c. &c. T. You have learnt to add 10 to every number ; if, now, it be required to add 11, what would you do ? P. First add 10, and then 1 more. T. How much is 15 more 11 ? P. 15 more 10, more 1, or 26. T. How much is 39 more 11? P. 39 more 10, more 1, or 50. T. And if it be required to add 1 2, what, then, would you do? P. First add 10, and then 2 more. T. How much is 57 more 12 ? P. 57 more 10, more 2, or 69. T. How much is 85 more 1 2 ? P. 85 more 10, more 2, or 97. ADDITION. 41 1\ If it be required to add 13, what would you do? P. First add 10, and then 3 more. T. Add 88 and 13. P. 88 more 13, are 88 more 10, more 3, or 101. T. If it be required to add 14, what would you do ? P. First add 10, and then 4 more. T. How much is 137 more 14? P. 137 more 10, more 4, or 151. To add the numbers, 15, 16, 17, 18, and 19, similar questions are asked. The above must be considered as a mere outline of the mode of pro- ceeding, not as being sufficient practice in the addition of these numbers. Each number may be taken for the subject of one lesson ; and before the pupils have recourse to the written exercises, verbal solutions must be given of questions refer- ring to that number. The pupils should give ques- tions to the class. Answers to the Exercises. Lesson I. Models. 57 + 10= 50 + 7 + 10= 67. 83+11= 83 + 10+1= 94. 149 + 12 = 149 + 10 + 2=161. 19=217 + ]0 + 9=236. 42 LESSONS ON NUMBER. To add 10. To add II. To add 12. Ans. 1. 59 Ans. 1. 94 Ans. 1. 61 2. 68 2. 68 2. 65 3. 86 3. 106 3. 90 4. 105 4. 114 4. 98 5. 111 5. 128 5. 105 6. 137 6. 165 6. 117 7. 163 7. 160 7. 170 8. 210 8. 187 8. 247 To add 13. To add 14. To add 15. Ans. 1. 61 Ans. 1. 51 Ans. 1. 79 2. 70 2. 72 2. 90 3. 99 3. 83 3. 102 4. 108 4. 87 4. 111 5. 130 5. 100 5. 132 6. 172 6. 111 6. 138 7. 201 7. 128 7. 153 8. 206 8. 150 8. 157 9. 218 9. 171 9. 166 10. 251 10. 293 10. 193 To add 16. Ans. 1. 2. 3. 74 91 105 Ans. 4. 5. 6. 114 133 140 Ans. 7. 8. 9. 10. 165 169 202 213 ADDITION. 43 To add 17. To add 18. Am. 1. 84 Am. 1. 77 2. 102 2. 94 3. 114 3. 112 4. 153 4. 131 5. 160 5. 155 6. 194 6. 176 7. 212 7. 147 8. 248 8. 204 9. 275 9. 315 10. 306 10. 336 To add 19. Am. 1 . 77 Ans. 5. 146 Ans. 9. 236 2. 106 6. 151 10. 278 3. 112 7. 194 11. 327 4. 133 8. 202 12. 408 From the mode hitherto pursued, the teacher will perceive how he should treat the addition of the numbers which follow. We have only to guard him against introducing any ready mode of solving the exercises ; as, for instance, that of arranging the numbers according to their local values. These exercises are only valuable, in as much as they are solved mentally. The work upon the slate must exhibit the process of the mind: thus, if it be 44 LESSONS ON NUMBER. required to add 79 and 48, the pupils proceed mentally, in the following manner : 79 more 48, are 79 more 40, more 8, which are 119 more 8; that is, 127. And upon the slate, the same has the following form : 79 + 48= 79 + 40 + 8 = 119 + 8=127. And this mode of showing the work upon the slate, it is advisable to continue, till the pupils have acquired sufficient practice in numeration ; after which, it may be abandoned, the pupils, simply writing down the answer, in the usual manner. LESSON II. To add 20, 21, 22 29. Answers to the Exercises. Ans.l. 57 Ans.9. 207 ,4^.17. 405 2. 66 10. 226 18. 412 3. 98 11. 274 19. 425 4. 111 12. 296 20. 440 5. 121 13. 318 21. 449 6. 138 14. 330 22. 66 7. 154 15. 363 23. 75 8. 165 16. 394 24. 84 ADDITION. 45 LESSON III. To add 30, 31,32 39. Am. 1. 88 Ans. 8. 271 Am. 15. 562 2. 118 9. 304 16. 93 3. 131 10. 386 17. 99 4. 150 11. 480 18. 102 5. 172 12. 500 19. 108 6. 209 13. 522 20. 114 7. 233 14. 540 &c. H 2 148 LESSONS ON NUMBER. T. How many thirds are there in 21 ? P. 6 g? ; because in 1 there are -f ; in 21, there- fore, there must be 21 times --, that is, 6 ^. T. What was the next sort of questions in our lesson on halves? P. Just the reverse. How many units are there T. How many units are there in 4 -/ ? P. 16-J-; because J- = 1 ; therefore, as often as 4 are contained in 4 T 9 , so many units there will be. Now f are contained in 4 T 9 16 times, and - more ; therefore 4 T 9 = 16. T. And what was the third sort of questions in halves ? P. How much is f of 18 ; f of 29, &c. T. How much, then, is % of 2 ? P. f ; because -|- of 1 = -g- ; therefore ^ of 2 must be 2 times i, that is, f . T. How much is of 4, 5, 6, &c. 100 ? P. 4. of 4 = H ; i of 5 = If; % of 6 = 2 ; and of 100 = 33. T 7 . How much is f of 2? P. 1^- ; because of 2 = f ; therefore f of 2 must be twice f ; that is, ^, or 1^. T. How much is f of 7, 14, 20, 50 ? P. Because of 7 = -J- = 2 ; therefore f of 7 = 2 x 24- = 4f . FRACTIONS. 149 Again, Because of 14 = V 4 = 4 t 5 therefore f of 14 = 2 x 4f- = 9. Also, Because of 20 = 2 T = Gf ; therefore f of 20 = 2 x 6f = 13^. And, Because *- of 50 = * T = 16f ; therefore f of 50 = ;2 x 16f "= 33i. Similar solutions are required for each of the questions given by the teacher and the pupils, after which the exercises in Part II. are taken up. Answers to the Exercises. Ans. 1. f means that 1 has been divided into 2 equal parts, and that one of these parts has been taken 5 times. And f means that 1 has been divided into 3 equal parts, and that one of these parts has been taken 5 times. 2. 7 = v ; 19f = V- 81 = ; *?i = 2 f 2 . 3. 2_s = 9^ an d 2_s = 14. 4_5 = 22J. = 39, and *y = 47, and i-f 1 = 70J. = 67, and 201 =100J. 150 LESSONS ON NUMBER. AmA. f of 19 = 12f ; f of 59 = 78f. f of 53 = 35-J- ; | of 48 = 80. | of 80 = 53; f of 50 = 100. f of 86 = 571- ; T. o f 81 = 189. | of 100 = 66f ; f of 82 = 218f. f of 126 = 84 ; of 100 = 300. LESSON III. Fourths. Teacher. Draw a straight line on your slates, and divide it into 4 equal parts. Pupils. 1 1 1 . T. What part of the whole line is one of these parts ? P. 1 fourth. T. And what are 2 parts ? P. 2 fourths. T. What are 3 parts? P. 3 fourths. T. And what are the 4 parts taken together? P. 4 fourths, or the whole line. T. If, now, we imagine the number 1 to be divided into 4 equal parts, what is each of the parts called ? P. One-fourth. T. Write in figures one-fourth. P. i. T. What are 2 parts called ? P. f- FRACTIONS. 151 T. What are 3 parts called ? P. f . T. What does the number 4 indicate ? P. That 1 has been divided into 4 equal parts. T. It is called the denominator. Thus, in the fraction f, what is the name of the parts into which 1 has been divided ? P. Thirds. T. Hence the number 3 names or denominates the parts into which 1 has been divided. And in the fraction f , what does 3 indicate ? P. That 3 of the 4 equal parts have been taken. T. That is, it shows the number of parts taken, and it is thence called the numerator. In -, which is the denominator, which the numerator ? P. 3 is the denominator, and 2 the numerator. T. How many fourths are there in 1,2, 3, 4, 5, &c ..... 100? P- %> t> v> y, 4 r- T. Our second sort of questions are the reverse of the above ; that is, how many units are there in P. H, If, 1|, 2, 2J.... 26. T. And our third sort of questions are, What is of 2, 3, 4 .... 17? p. f, i, i....4i. T. What is f of 37 ? P. i of 37 = = 9i ; and f of 37 = 2 x 9J = 18|. 152 LESSONS ON NUMBER. T. How much is | of 23 ? P. i of 23 = 2_p = 53 . and | of 23 = 3x53 = 17J. Answers to the Exercises. Ans. 1. The denominator of a fraction indicates how many equal parts 1 has been divided into ; and the numerator shows how many of the equal parts have been taken. 2. 53=; 73 = ; i9t = V; 3*3 = if*; 113J = 4 | 3 5 239 I = H 9 271i = 10 T 85 . 3. v= 8i; V 7 = 6*; V= 4t. V = 43 5 V =28f; y =21f. 166|. 4. f of 17 = Hi-; e O f 33 = 49^ | of 17 = 12|; i of 25 = 43|. f of 82 = 54f ; f of 136 = 272. | of 82 = 61f ; f of 116 = 261. J of 96 = 72; u>of 44 = 110. f of 71 =88|; V of 29 = 79|. FRACTIONS. 153 LESSON IV. Fifths, Sixths, Sevenths. If a straight line be divided into 5, 6, 7 equal parts, the pupils will arrive at the idea of 5ths, 6ths, 7ths. A process quite analogous to that pursued in the previous lessons, is then applied to number 1. The following are instances of these sorts of questions, referred to in Lessons I., II., III. 1. Reduce 1, 2, 3, 4 98, 99, 100 to 5ths. .. 1,2,3,4 99, 100 to 6ths. .. 1,2,3,4 99, 100 to 7ths. 2. Reduce to units f, , T 8 T 9 / 1 f. " 78 9 99100 ^> W) ~5 6 9 6" .. f, *, V V. *r- 3. How much is f of 27 ; f of 39 ; f of 63, &c.? .. f of 31; f of 48; | of 55, &c.? .. $ of 17; f of 30; A of 43; f of 100, &c. ? Answers to the Exercises. Ans.l. 2 | 5 ; 2 | 5 ; 5 f * ; 6 | 5 ; 10 -/ ; 1S -/ S , 2. 17f ; lOf ; 19f ; 22| ; 33$ ; 43f ; 68f ; ; 112J; 135*. H 5 154 LESSONS ON NUMBER. Ans. 3. $ of 53 = 42f ; f of 216 = 129$ ; f of 89 = 53f ; f of 331 = 132f ; I of 117 = 46f ; | of 459 = 367 ; f of 189 = 15l. 4 . 034; 582; 690. I OJ 4 . 1 4_,2 8 . 27^54, 5. 15$; 24f; 30f ; 36^; 57^; 76; 96; 113; 148f. 6. ^ of 86 = 14f ; f of 187 = 155f ; f of 93 = 31; I of 44= 51f; fofll5 = 57f; fof 80=106f; f of 139 = 92|, 7^ 6 5.1 819. 1 3_0 2 1 4_9 8 . 2 1_2 8 . 2 9 5 . 8. 7f; 12$; 13|; 214; 27; 31; 35^; 46$ ; 69$ ; 77 ; 97$. 9. $of 86= 12$; fof 200 =171$; $of 94= 26|; fof 319 = 273$; f of 112 = 48 ; f of 480 = 342f ; $ of 186 = 106$ ; $ of 590 = 337$ ; fof 190=135$; $ of 640 = 274$. LESSON V. Eight/is, Ninths, Tenths. Answers to the Exercises. .}. H S ; 6 f 8 ; " 6 ; 'V 8 ; 14 16 ; 'V 4 ; 2. 11$; 18; 29* ; 39| ; 56f; 64|; 84; 97i; 106; FRACTIONS. 155 3. | of 49 = 6i ; of 156 = 97$ ; fof 73 = 18f; of 200 = 150; of 100 = 37$ ; | of 346 = 302-f ; ! of 115 = 57$. 4. *4. $5; i^ 7 ; 19 5 44 ; 3 V 9 - 5. 4$ ; 12* ; 20^ ; 24^ ; 39J. 6. of 86= 9|; | of 134 = 74$; f of 144 = 32 ; f of 242 = 161| ; f of 63 = 21 ; $ of 313 = 243| ; $ of 1 15 = 51| ; of 516 = 458$. 7. w > i n > 2 H ; 3 -B- 8. . 9. T V of 87 = 8^; T % of 115 = 23; A of 186= 55 T ^; T % of 217= 8ft&; T \ of 316 = 158; A of 419 = 251 T %; T V of 528 = 369 T Reduce | to 28ths. P. | = 2. -i., because 1 = f f ; therefore J = J of f f = ^, and $ = 3 x T 7 -g = |-|-. T. Find 6 different expressions for j. P3 6- 9 12 15 18 21 f "8 TS T6" ~2 U ~ 2- T 2"8' I 7 . In what numbers can 5ths be expressed ? P. In lOths, 15ths, 20ths, &c. ; in all numbers which are multiples of 5. T. Reduce $ to 25ths. P. = ^ because 1 = f f ; therefore -J- = of If = * T. Reduce f to GOths. ^- * = ** because 1 = f ; therefore = of f % = Jf , and f = 4 x B If; T. Find 6 different expressions for f . * * = T 6 o = T% = M = if = if = H- A similar mode of proceeding is to be followed as to the reduction of 6ths, 7ths, 8ths, &c. to other expressions. The teacher will have observed that the use of the line has been abandoned for a sort of systematic reasoning ; yet, whenever the idea is FRACTIONS* 161 not quite clear in the pupil's mind, it is recom- mended to have recourse to the senses, by requir- ing a line to be divided, according as the question may require. Answers to the Exercises. Halves. Am. 1. } = T T = = ft = If = T T V 2- * = = #= Vtf = V-4 1 = 111- Thirds. 3- l = A = M = H = ff = Tr- 4- I = M = 41 = HI = 5- 4 = ^ = 11 = 11 = 6- l = = = = _3. 5 4^ JL.2.JL 4 .Q.2. 3" 1JO 4 8"5" 670* 9- = = = Sevenths. 10. 4 = M = -i ff = ?& = ? = M = T 3 A = T 4 /8 = 162 LESSONS ON NUMBER 4 = M = A\ = T'A = 4 = M = T and i = A; or, &c. T. How will you proceed in order to reduce \ and \ to other fractions having the same denomi- nator ? P. First find the several expressions for J and J, and then choose those which are the same. T. Reduce \ and J to fractions having the same denominator, or, as it is usual to say, to a common denominator. 164 LESSONS ON NUMBER. -P- 1=| = t = | = T 5 tr = T f V. &c. i = I = A = T 4 o> &C. Hence, the same denominations for J and J are, 4ths, 8ths, 12ths, 16ths, &c. Or, i = f = = ^ = T 8 a> &c- and J = = t V = T 4 &c. T. What are the numbers 4, 8, 12, 16, &c. of the denominators 2 and 4 ? P. They are common multiples of 2 and 4. T. Hence, if 2 fractions of different denomina- tors are to be reduced to a common denominator, what must be done? P. We must find the common multiples of the different denominators, and reduce the fractions to these denominations. * T. Reduce J and f to a common denominator. P. The common multiples of 2 and 3 are, 6, 12, 18, 24, &c. and 1 = 1 = ^ = 11 = 11,^. T. Reduce \ and 1 to a common denominator. P. The common multiples of 2 and 5 are, 10, 20, 30, 40, 8tc. Hence \ = ^ = ft = ^ = f g, &c. and = = = = . &c - * See Chap. V., Lesson II. FRACTIONS. 165 T. Reduce f and f to a common denomi- nator. P. The common multiples of 3 and 4 are, 12, 24, 36, 48, 60, &c. Hence = A = if = fa = ft = *** &c. and | = A> it = fa 1 = If = H, &*- T. Which of the several common denominators is the least ? P. 12. T. Hence, if 2 fractions are to be reduced to the least common denominator, what must be done ? P. Find the least common multiple of their denominators, and reduce the fractions to that denomination. T. Reduce f and f to the least common denomi- nator. P. The least common multiple of 8 and 5 is 40; hence f = if, and $ = . T. Reduce \, 4, and J to the least common denominator. P. The least common multiple of 2, 3, and 4 is 12; hence J = T 6 Y ; -J- = T V 5 aud i = A- Pupils and teacher give questions to the class, after which follow the exercises in Part II. 166 LESSONS ON NUMBER. Answers to the Exercises. Am. 1. | =fj; * =** 2. I =1-1; * =if- 3. f =tf; i =if. 4. 4 =11; t =M- 5. f = M ; T'O = M- 6- T 9 o = ** ; T V = f % 7. * = 1 1 ; A = U- 8. * =i$; * =il- 9. I =A; f =ff 10. H = M ; if = If- 11 30 40 45 48 50 42 55 5 JJ - 6~0> ^> 3~^> ^^ ^"0"? 6"^' S~0 ^" **" 2"T "2T? 2"T? "2T^ 2"T> 2"T* *, iM. m- 15. fB, fW, H*. Iff- is. m, it*. IM. m- 3. ADDITION OF FRACTIONS. LESSON I. Addition of Fractions which have the same Denominator. Teacher. Tell me what we have been learning concerning fractions. FRACTIONS. 167 Pupils. We have learnt, 1st. What halves, thirds, fourths, &c. are; 2nd. To reduce units to fractions ; 3rd. To reduce fractions to units; 4th. To reduce a fraction to other denomina- tions ; 5th. To reduce several fractions to the least common denominator. It is of great use to put, from time to time, questions similar to the above to the pupils, and require them to illustrate each case by examples. Teacher. All that now remains to be learned re- specting fractions is, how to add them; to sub- tract one from another; to multiply and to divide them. We will begin with addition of fractions. Give a question which you think very easy to answer. Pupils. Add j + J. T. How much is J + \ ? P. f , or 1. T. Give some other easy questions. P. AddJ+f. T. How much is j+f ? P. f, or 1J. T. Addf + f. P. f x-f = y = 7. T. P. f. 168 LESSONS ON NUMBER. T. Addf + f. P. |. or 1|. T. Add J + |. P- 3 + f = f = 2. T. What kind of fractions have we been add- ing? P. Fractions having the same denominator. T. Add f + f. P T _ 1 2 * 5" X T- Similar questions are to be given in the addition of 6ths, 7ths, 8ths, &c. LESSON II. Addition of Fractions having different Denominators. Teacher. If now we have to add fractions of different denominators, suppose |+^, what will you do ? Pupils. First reduce \ and ^ to a common deno- minator, and then add them ; thus : i+f= m = i- Teacher. Add J+f. p. i+i = *+* = j = H. T. Add i + i. -P. i + J = f + J = i- T. Add J + j. p. i + 2 = f + | = |= ii. T. Addi+f FRACTIONS. 169 r. Add j+$. -P. J+* = f +* = *=* if- !/'. There is something to be remarked here. The answer is If, do you know of another expres- sion for f ? P. Yes ^, or -% , or ? 8 T , 8cc. T. True ; but do you know of another expres- sion for below 12ths ? P. Yes, *. T. Now, it is usual to express a fraction by the least expression possible ; or, as it is called, to reduce it to its lowest term. Hence f- reduced to its lowest term is - P.I. T. Add i+. T. Can ^i be reduced to a lower term ? P. No. T. Can % be reduced ? P. Yes, if = f . jf. In our next lesson we will learn what frac- tions may be reduced to lower terms, and what may not. Teacher and pupils, as usual, give similar ques- tions to the class, after which the exercises in Part II. are taken up. i 170 LESSONS ON NUMBER. Answers to the Exercises. Ans. 1. 1-^-. Ans. 13. Hi = Hi- 2. Iffr. 14. IM- 3. Hg. 15. Hi 4. Iff- 16. Iff. 5. 1J.1-. 17. liHr- 6. 14*. 18. Ifi-. 7. H&' 19. 1|-|.. 8. 1-JLSj. 20. 1|| = H.I.. 9. Hfl- 21. l|f- 10. li- 22. HI = iff- 11. Iff. 23. HI = 1 H- 12. Iff. LESSON III. Teacher. State the different expressions for J. P WF 7 5 . 1 = f = f = A = ^ = T 6_, &c . T. Since J = T 6 ^, what must be done to bring T 6 ^ back again to its lowest denomination J ? P. Divide its numerator and its denominator by 6. T. State the different expressions for ^. -P- i = f = -| = A = T 5 ^Scc. T 7 . Since ^ = T 5 T , what must be done to reduce ^j to its lowest term |- ? P. Divide its numerator and its denominator by 5, FRACTIONS. 171 T. Let us now consider the fraction -J-|, for instance, and see by what number its numerator and denominator must be divided, in order to reduce \ to its lowest term. P. They must be divided by 2, and then -J-| will be reduced to -J-. T. Can you reduce ^ to a lower term? P. No, because there is no number by which both 7 and 9 are divisible. T. Reduce if to its lowest term. P. 18 and 24 are both divisible by 2, therefore |f = T 9 2- ; and 9 and 12 are divisible by 3, there- fore T 9 = | ; and hence |~f = f . T. In general, then, when can a fraction be re- duced to lower terms, and when not? JP. It can be reduced when both numerator and denominator are divisible by some one number; and it cannot be reduced when numerator and denominator are not both divisible by the same number. T. Recollect then to reduce the answers you obtain by adding, subtracting, multiplying, or dividing fractions, always to its lowest term. Can you tell some reason why it is usual to do so ? P. Because in the fraction T 9 ^, which, when re- duced, is j ; it is easier to imagine the number 1 to be divided into 4 equal parts, and 3 of these parts taken, than to imagine 1 divided into 12 equal parts, and 9 of these taken. i 2 172 LESSONS ON NUMBER. T. We will now add 3 or more fractions, [writ- ing upon the school-slate,] Find the sum of J+|- + J. P. The least common denominator is 12, and J+*+ i = * + A+ A = if = HV T. Add J + f + |. P- J+f +1 = A+A+A = ft = Hi- T. Add J + i + f P. T. Add P. j + i +A = T. Add P. P. 1| + If = Iff + 1ft = 2^ = Exercises. Am. 1. 2f. 4ws. 10. 2. 2H- 11- 3. Hf. 12. 4. 2 T ^. 13. 5. 214- . 14. 6. 2if. 15. 7. 2#. 16. 8. 6 T V 17. 9. 3. 18. FRACtlOiNS. 173 ^ 4. SUBTRACTION OF FRACTIONS. This lesson is, in all respects, analogous to addition of fractions, and does not involve any new principle ; a few questions relating to the several progressive steps, will therefore suffice to show the mode of proceeding. 1. Subtracting a Fraction from Integers. Questions. From 1 take J. . . 2 take . . . 3 take f . ..17 take If. 8cc. 2. Subtracting a Fraction from a Fraction, (a.) How much is J less \ ? f? (6.) What is the difference between 1 and ? -=$- = . .. J and i? J-J=: f- i = J. |. and I? |-| = T 8_- 1 P = = TV .. I and A? 4-6 =f B^O =T I_. 174 LESSONS ON NUMBER. 3. Finally, subtracting Integers and Fractions from Integers and Fractions. From 2J take If. Solution. 2J~ If = f-f = V - V = f From 3 take 2$. Solution. 3-2 = y- V = W-W = it- It is to be observed, that mental solutions are required, and therefore no questions are to be given involving high numbers, the object being that of unfolding principles, and preparing for the rule, and care being necessary not to perplex the minds of the class. Each question may be written on the large school-slate, and the pupils required to find the answer mentally, each lifting up his hand as soon as he has found the answer. Answers to the Exercises. 1. Fractions from Integers. Am. \. 16 T V Arts. 6. 42*, 2. 18f|. 7. 24|f. 3. 34J. 8. 46i|. 4. 7J. 9. 58ff. 5. 18f 10. 66ff. FRACTIONS. 175 2. Fractions from Fractions. Ans.l. if Ans.ll. f*. 2. ft. 12. tf. 3. if. 13. A- 4. ,& 14. ff. 5. 1. 15. ii. 6. if- 16- A- 7. 0. 17. T \=f 8. T V 18. 0. 9. ^%. 19. 10. . . 20. 3. Integers and Fractions from the same* Ans.l. 1. Ans.6. 2fi. 2. IH- 7. f. 3.. 2 T V 8. 1J. 4. Hf. 9. 5. 3^. 10. 5. MULTIPLICATION OF FRACTIONS. The order in which the successive steps of this subject present themselves are : 1st. To multiply a fraction by integers ; 2nd. To multiply integers by fractions; 3rd. To multiply a fraction by fractions; 4th. To multiply integers and fractions by the same. 17-6 LESSONS ON NUMBER. On the first of these sections no remark is re- quired to be made, since pupils who have pro- ceeded thus far, find no difficulty in multiplying f , for instance, by 2, 3, 4, &c. since it is only to take f twice, 3 times, 4 times. The second section might, apparently, be dis- pensed with, since the results are the same, whether 3 be multiplied by ^, or ^ be multiplied by 3. But this principle, clear as it is in the multiplication of integers, is far from being satisfactory when ap- plied to fractions. A child will not hesitate readily to admit that |, multiplied by 15, gives \, or 10 ; but it will not immediately perceive that 15, mul- tiplied by -, must be 10.* The question may be presented thus : 1. What does it mean to multiply a number by another number ? AJIS. To take a number a certain number of times. 2. What is the result if a number be multiplied by 1? Arts, The number itself. 3. If, then, a number be multiplied by less than 1, (by a part of 1,) what must be the result ? Ans. Less than the number. 4. If a number be multiplied by J, what must be the result ? Ans. One-half of the number. FRACTIONS. 177 5. If multiplied by ? Ans. One-third of the number. 6. If by f? Ans. One-third of the number taken twice. 7. Hence, what is the meaning, and what is the result, of 15 multiplied by f ? Ans. To multiply 15 by f , signifies to take ^ of 15 twice, which evidently is 10. The notion, then, to be clearly formed is this, that any number whatever, multiplied by another less than 1, must give a result less than the number which is to be multiplied (the multiplicand.) Sections 3 and 4 are merely an extension of the same principle. LESSON I. Fractions by Integers. Teacher. What does it mean to multiply? Pupils. To take a number a certain number of times. T. What does it mean to multiply J by 1, 2, 3, 4, &c. P. To take \ once, twice, 3 times, 4 times, &c. T. How much is \ multiplied by 17 ? P. V,or8j. T. What does it mean to multiply by 1, 2, 3, 4, 8cc. P, To take |- once, twice, 3 times, 4 times, &c i 5 178 LESSONS ON T. Multiply-!- by 19. P. -1x19= V 9 = 6. T. Multiply!- by 19. P. fxl9 = y = 12f. T. In general, then, what does it mean to mul- tiplyf,f,f,&c.byl,2,3,4,&c.? P. To take f, f , , &c. once, twice, 3 times, 4 times, &c. T. How much is -| multiplied by 8 ? P. 8 times f, or 2 T 4 which are 3-f. T. How much is -J x 15 ? P. 15 times f , or V which are 10, LESSON II. Integers by Fractions. Teacher. What is the result (product) if a num- ber be multiplied by 1 ? Pupils. The number itself. T. What is the product if a number be multi- plied by a number greater than 1 ? P. More than the number* T. What, then, must be the product if a num- ber be multiplied by a number which is less than 1 ? P. It must be less than the number. T. Name some numbers which are less than I ? P> \> *, I, i, *, 8cc- T. If, then, a number be multiplid by J, or by i, J, &c. ; the product must in each case be FRACTIONS. 179 P. Less than the number. T. What does it mean to multiply by 1 ? P. To take a number once. T. Hence, what does it mean to multiply a num- ber by \ ? P. To take it \ times, or to take \ of it. T. What does it mean to multiply 1, 2, 3, 4, &c. by J? P. To take J of 1,2, 3,4, &c. T. How much, then, is 1, 2, 3, 4, &c. multiplied by, 1 ? P. }, I, f , f , &c. T. How much is 17 multiplied by \ ? P. V,or8i T. What does it mean to multiply by f ? P. To take \ of a number twice. T. How much is 12 multiplied by f ? P. i of 18 taken twice ; i of 18 = 9, which, taken twice, = 18. T. What have you to remark, if a number is to be multiplied by f ? P. It is the same as multiplying it by 1, since 1 = 1. T. What does it mean to multiply by 4 ? P. To take a number - times, or to take ^ of it. T. And what does it mean to multiply by f ? P. To take ^ of a number twice. T. Multiply 17 by f. 180 LESSONS ON NUMBEK, P. 17 xf is^-of 17x2; ^ of 17 = y, which, taken twice, T. What does it mean to multiply by J, ^, , 4* &c.? P. To take 4, , , ^ of a number. T 7 . And what does it mean to multiply by | ? P. To take J of a number 3 times. T. Multiply 9 by j. P. 9 x f is 4 of 9 x 3 ; 4 of 9 = f , which x 3 = V = 6 f T. What does it mean to multiply by 4? P. To take \ of a number 6 times. T. Are you able to multiply a whole number by a fraction ? P. Yes ; we have learnt it in our first lessons on fractions. T. What kind of questions were these? P. To take f, j, ^, &c. of a number; and this is the same as to multiply a number by f, or f , LESSON III. Fractions by Fractions. Tetocher. We must now learn to multiply a frac- tion by a fraction ; and we will begin with ascer- taining what it means to multiply ^, for instance, by ^. You know the meaning and the result if ^ be multiplied by ! FRACTIONS. 181 Pupils. Yes, it means to take | once, which is |. T. What, then, does it mean to multiply J byj? P. To take i of J. T. How much is that ? One or two of the pupils, perhaps, will answer this question correctly, the majority not. Recourse must then be had to ocular demonstration. T. Draw a straight line ; divide it into halves, each half again into halves ; now tell me what part of the whole line of one of these halves is \ ? P. One-fourth of the line. T. Apply the same reasoning to the number |, and tell me what J of $ is ? P. i. T. Hence how much is \ x \ ? P. \ of J, or J. T. What does it mean to multiply 4- by \ ? P. To take J of . T. You may ascertain this by drawing a line ; how will you proceed ? P. Divide a line first into thirds, each third then into halves, and see what part \ of is of the whole line; it is J of it, T, How much then is -J- x \ ? P. J of $, or -i, T. And how much is f x \ ? P. J of f , or i-. 182 LESSONS ON NUMBER. T. What does it mean to multiply J, , , i, &c. byj? P. To take J of J, of , of , of |, &c. T. Hence if you wish to learn how to multiply a fraction by J, you must be able to ascertain readily how much \ of \> of , &c. is. Need you always take a line and actually divide it? P. No, we can imagine it. T. Well, then, ascertain either by drawing a line and dividing it, or by supposing it divided, how much \ is of J, of ^, of f, J, J, , f, f , f, &c. P. Must be able to draw up the following results : i = K I = TV f = 4 = i- TV = irV f = i- T 9 o = A- I = f ft = M- T. A little reasoning will save you a great deal of trouble. For instance, how much is \ of % 1 P. TV' T. How much, then, is of f , f, $ , | ? P. 2 x ^, 3 x T V, 4 x T V, 7 x T V, or -&, T ^, T\> TV' T. And if you know how much \ of is, can you tell me how much f , f , ^ of ^ is ? FRACTIONS. 183 P. Yes; for i of i = ^5 fofi = 3x T 1 = T V i of | = 5x^=^ = 1. iofi = 7x T V = T V .7'. Hence, how much is f multiplied by f ? *"*<*--Ari i. of f = 3x 1 V = T 3 cr; and | of f = 4xtV = l = f =lf ' jf 7 . How much is - x |- ? ^ iofi = T V; | of | = 8x T V = T 1 V = l; and | of -I == 5 x | = % = 2f ; therefore A sufficient number of questions relating to the multiplication of fractions by halves, ought to be given before proceeding further ; and it must be remarked, that most children will soon discover the rule, viz. to multiply numerator by numerator, and denominator by denominator ; but since it is not the object of this treatise to enter upon rules, but merely to prepare for them, the teacher ought frequently to require of his pupils to give an ac- count how they have obtained the result. From the above, the mode of proceeding as to the multiplication by --, J, , &c., may be antici- pated, and a short outline will be sufficient. 184 LESSONS ON NUMBER. The pupils must ascertain that . . i of i = i. i of i = -i. * = fr- * = TV i = A- i = TV i = TV i = ?V &c. &c. i of i = T V i of 1 = T V i- = TV i = TV _ 25' &c. &c. This done, and committed to memory, is all that is necessary. Teacher. What does it mean to multiply by ? Pupils. To take ^ of a number. T. What does it mean to multiply \ by - ? P. To take 4 of |. T. How much is that ? p. *. T. How much is 4 x|? P. i-of 4; iof | = ^ T ; ^d ^- of -f- = 4 x ^j = -2 4 T ; therefore 4x^ = 7 V 7 1 . How much is f x | ? FRACTIONS. 186 P. f of f . Now i of = ^V ; and | of f = 8 x ^ = -fo ; therefore f of -f = 2 x 2 B T = if- Hence $ x f = -|f T. How much is -| x | ? P. f of f Nowiof i = ^; therefore J of | = 7 x -^ = 3% ;' and f of | = 3 x ^ = f. Am* Answers to the Exercises. 1. Fractions by Integers. Ans. 1. 3. ^w. 11. 27-f. 2. 5. 12. 41*. 3. 7. 13. 73. 4. 12. 14. 78. 5. 317. 15. 29. 6. 4|. 16. 52. 7. 13f. 17. 91. 8. 25|. 18. 112. 9. 40f. 19. 142. 10. 32^. 20. 311. 2. Integers by Fractions. Ans. I. 31*. Ans. 6. 3434. 2. 64f 7. 78-f. 3. 93|. 8. 249|. 4. 88f. 9. 24. 5. 270. 10. 91*. 186 LESSONS ON NUMBER. 3. Fractions fty Fractions. Ans. 1. * Ans. 9. 2. H* 10. 3. M- 11. 4. if- 12. 5. T 3 ~7> CCC * T. Now, since 1 is contained in I, once, will 1 be contained as often in a number which is less than 1 ? P. No; less times. T. Hence, how often is 1 contained in \ ? P. Only \ times. T. How much then is |-r-l ? P. i. T. And how often is 2 contained in \ ? 190 LESSONS ON NUMBER. P. Only ! times of what 1 is contained in |, that is \ times. T. How much then is |-r-2? ^i- T. And how much is divided by 3, 4, 5, 6, 7, &c. ? *, *, T. How often is 1 contained in ^ ? P. Only % times. T. How much then is -g-r-1 ? p.*, T. How often is 2 x 1, or 2 contained in ^? jP. -^ of ^ times, ^ times. T. Hence ~3 is? P r f T. How much is divided by 4, 5, 6, &c. ? ^' TV, TV, T^ &C. T. And how much is f divided by 1 ? P. f . T. Why? P. Because |~1 = i, therefore f-r-1 = 2 x $ = 2f . T. How muchisf-r-2? P. ^ ; because f -f- 1 = f , therefore f -r2 = \ of * = i- T. Howmuchisi-T-l? ^i- T. How much is f ~2 ? P. |; because |~1 = f, therefore f-f-2 = \ of = . FRACTIONS. 191 The pupils must continue to give similar solu- tions for each question, which the teacher may give. These questions may be solved in another way. Dividing by 2 signifies to take J of a number. o -3- 4 &c. &c. Hence, -|-r-2 = \ of \ = \. i-f-3 = i of \ =: i. &c. H-5 = * f * = T V t-5-9 = i of i = T V &c. Still the first mode of considering division would be preferable. 2. Integers by Fractions. Teacher. How often is 1 contained in 1, 2, 3, 4, &c. ? Pupik. Once, twice, 3 times, 4 times, 8cc. T. How often then is \ contained in 1, 2, 3, 4, &c. ? P. \ is contained in 1, twice ; 2, 4 times ; 3,6 times ; 4, 8 times. 192 LESSONS ON NUMBER. T. How much then is 1---^, 2-f-^, 3-r-^, 4~| ? P. 2, 4, 6, 8. T. How often is ^ contained in 1, 2, 3, 4, &c. ? P. is contained in 1, 3 times ; 2, 6 times ; 3,9 times ; 4, 12 times ; &c. T. How much then is 1-r-i, 2-f-, 3-f-|, 4-r|? P. 3, 6, 9, 12. T. Since ^ is contained in 1, 3 times, how often must f be contained in 1 ? P. Only -| as often as ^ is contained in 1, that is, \ of 3, or f times. r. Then how much is l-hf ? % P. |, or 1|. T. How much is 2-r-f ? P. 3; because 2^- = 6, therefore 2-f-f = | of 6 = 3. T. How much is 3-r-f ? P. f or 4^ ; because 3---^- = 9; therefore 3-j-f. = i of 9 = 4|. T. How much is 1, 2, 3, 4, &c. divided byi? P. 1^-i = 4; 2-5-i =8; 3-=-i = 12; 4^-i = 16, &c. T. How mucH is 1-r-f ? P. f or 1|- ; because I-T--T = 4, therefore l-:-f = 4 of 4 = 4 = H. FRACTIONS. 193 T. How much is 7~f ? P. 2 3- 8 or 9; because 7-^-1- = 28, therefore 7-f-t = * of 28 = V 8 = Si- Similar solutions are required for each of the questions which the teacher and pupils may give to the class. The principle upon which these solutions depend can be made very obvious to the senses ; for, it is clear, that the smaller the mea- sure the greater the result obtained by applying it to an object. Suppose a foot measure to be ap- plied to the length of a table 10 times ; it is not necessary actually to apply a -J foot measure, or , f , i, |- , &c. of a foot measure ; for since a 1 foot measures the object 10 times, a \ foot must evi- dently measure it 2 x 10, that is 20 times ; i of a foot 3 x 10, that is 30 times ; but a f foot measure only \ of 30, that is 15 times ; a -J- foot measure 4 x 10, that is 40 times ; but a f foot measure only \ of 40, or 4 T , or 13^ times. The intelligence of children of 8 or 9 years old readily seizes these truths, and little effort is then required to apply a similar train of reasoning to abstract numbers. Whenever a child seems perplexed by a question of the kind, it will surely arrive at the true result, if accustomed thus to reason. Suppose the question; divide 3 by -, the child begins from what he is certain to be true, viz. : 1-H = 9; therefore 3-r- == 3 x 9 = 27 ; and hence 3-f-J- = \ of 27 = V = 3f 194 LESSONS ON NUMBER. Again, divide 7 by -|. Solution. Because 1-:-^ = 9 ; therefore 7-f-J- = 7 x 9 = 63 ; and therefore 7-f-f = -J- of 63 = ^ = 7. These solutions the pupils are required to per- form mentally ; but the written exercises should be solved as shown above. Answers to the Exercises. 1 . Fractions by Integers. Ans. 1. i. Ans. 9. ^V 2. T v 10. -AV 3. A- 11- i- 4. A. 12 - *T- 5. A- l3 ' A- 6. f. 14. T T . 7. A- 15. 8. 2. Integers by Fractions. Ans. \. 8. 4m. 7. 25| 2. 8|. 8. 4f 3. 18f. 9. 4. lOf 10. 8. 5. 26f. 11. 7. 6. 16|. 12. 7. FRACTIONS. 195 Am. 13. 8. Ans. 17. 13. 14. 10. 18. 4. 15. 11. 19. 5. 16. 12. 20. 9. 3, Fractions by Fractions. The solutions of questions of this section are only an extension of the principles laid down in the former section to fractional numbers. For, since it is known that \ is contained in 1 twice, it necessarily follows that \ is contained in -J- of 1 only \ of twice, that is, once ; or, because 1 -r- = 2 ; therefore -J-r-i = \ of 2 = 1. Again, because it is known that ^ is contained in 1, 3 times; it follows, that -- is contained in | only ^ of 3 times, that is, f times ; in other words, because 1-r-^- = 3 ; therefore i-H = i of 3 = | = 1|. And so on with other questions of the kind. Teacher. How often is \ contained in 1 ? Pupils. Twice. T. Hence, how often must \ be contained in ^ of 1, or in i? P. Only \ of what it is contained in 1, that is once. K 2 196 LESSONS ON NUMBER. T. How much then is %-r-% ? P. 1. . T. Again, how often is \ contained in 1 ? P. Twice. T. Then how often must % be contained in the 3d part of 1, or in -J-? P. Only the 3d part of what it is contained in 1, that is f- times. T. How much then is -J-f-J ? P. J-; because 1-h^ = 2 ; therefore i-r-i = i of 2 = f. T 7 . Also, because 1-=--^ = 2; therefore -|~|- is P. i of 2, or f, or |. T 7 . And, because 1-f-i = 2 ; therefore ^-r-^ is P. i of 2 = f. T 7 . And, because 1-f-i = 2; therefore ^-r-^ is P. of 2 = f = i. T. How much is i, ^, T ^, &c. -H ? P. Because l-r-| = 2; therefore i-r-i = i of 2 = f = i Again, because 1-f-^ = 2 ; therefore ^-7-^ = of 2 = f ; and because 1-f-^ = 2 ; therefore ^-f-i = T V of 2 = -ft = i- T. How much is f -T-i? FRACTIONS. 197 P. Because 1 \ = 2; therefore = of 2 = f ; and therefore f -J- = 2 x f = |- = \\. T. Divide f , f, f , f , 8cc. by J. P. Because 1-f-J = 2; therefore i-=- J = O f2= = J; and therefore f.^i = 3xj=$ = lj. Also, because 1-f- J = 2 ; therefore *-r- \ = f of 2 = f ; and*-:-i=4xf = f=lf. Again, because 1-rg == 2 ; therefore -=- J =iof2 = f = |; andf~J = 5x^ = 4-= If. Also, because 1-i-J = 2 ; therefore -^-H 1 = \ of 2 = f ; and 4-r- J = 6 x f = V 2 = H- T. How muchisi-7-^-? P. Because l-h| = 3 ; therefore $+$ = ^ of 3 = f = 1|, T 7 . And how much is -J-T-f? P. Because I-:-*- = 3; therefore -|-f-^- = \ of 3 = f ; and therefore Y-rf = \ of f = f . In this solution, the teacher has to bring forcibly before the mind of his pupils the prin- ciple, that, " if a number be contained in another 198 LESSONS ON NUMBER. number 6 times, twice the former can only be con- tained in | of 6, that is, 3 times." And 3 x times the former, only -J- of 6, that is, twice, 4 x times the former, only -J- of 6, that is, = | times. &c. And hence, because it has been ascertained that i-r-g- = f ; therefore i-~2xi=ioff = f. Teacher. How much is ^-r-f ? Pupils. Because 1-r-^- = 3 ; therefore -J-r-J- = of 3 = 1 ; and therefore i-f-f = J of 1 = \. T. How much is i~| ? P. Because l-f-|- = 3; therefore -?- = -J. of 3 = f ; T 7 . How much is f I? P. Because 1-f-^ = 3; therefore %+% = i of 3 = f ; . andf+i = 3 xf = f; and therefore f H-| = \ of f = % = 1-^. The pupils must be able to give similar solutions for each question. FRACTIONS. 199 The principle remains unaltered, whatever the question may be ; for, let it be required to divide i by*. Because 1~^ = 9 ; therefore ^-f~ = \ of 9 = % ; and -f -f-^ = 3 x $ = V 5 and therefore f -f-f = of V = f . Again, to divide -| by f . Because 1-f-^ = 5 ; therefore I-7--J- = ^ of 5 = |- ; hence f -^ = 5 x f = \ 5 ; and therefore |~f = -J- of 2 -g 5 = f f = 1 V Answers to the Exercises. Am. 1. \\. Ans. 13. -f-^. 2. 1 T V 14. -|f- 3. 4. 15. 3||. 4. If. 16. fi. Pi 3 17 91 3 tj. -T-. If. ^"5~S"' 6. l^p 18. f. 7. H. 19. f. 8. Iff. 20. |. 9. |f. 21. If 10. fi. 22. |. 11. lil- 23. f. 12. l|i. 24. |. 200 LESSONS ON NUMBER. 4. Integers and Fractions by the same. Teacher. What are the kinds of questions you have now learnt to perform in division of frac- tions ? Pupils. 1. To divide fractions by integers; 2. To divide integers by fractions; 3. To divide fractions by fractions. T. What do you think now remains to be learnt ? P. To divide integers and fractions by integers and fractions. T. Give a question of the kind, and let us see if you are able to answer it. P. Divide 1| by 1*-. T. Which is the greatest of these numbers? p. H. T. Can you tell whether the answer (quotient) is to be less than 1, or more than 1 ? P. Tt must be more than 1, because 1^ is less than 1^ ; it must, therefore, be contained more than once in 1^. T. Well, then, how much is l^-r-l^? P. 1^-f-l^ is the same as f -f-f- ; and because 1-r-^ = 3, therefore H4- = of3 = $, and f -r-^- = 3 x f = | ; and therefore f-f = ioff = |=H. Hence l^-r-l^= l. FRACTIONS. 201 T. Divide l by 1|. Is the quotient to be more or less than 1 ? P. Less than 1, because \\ is more than l, and cannot, therefore, be contained once in 1-J-. which is -f. T. Hence you can always tell beforehand, whe- ther the quotient is to be less or more than 1 ; and I advise you to ascertain that before you begin finding the answer, as it will show you whether your answer approaches the truth or not. Divide 3| by 4^. P. The quotient must be less than 1. 3i-r4tf-ry- Now 1 -r- = 5 ; therefore l -f -f- * = = f . .Aws. The pupils must give a similar account for each of the questions given to them ; and in the exer- cises of Part II., write out, as shown above, the process by which they have obtained these re- sults. Answers to the Exercises. Ans. 1. J-. Ans. 7. . 2. If. 8. -&. 3. 1-fr. 9. 5f 4. |$. 10. 7. 5. ii. 11. 4f. 6. H. 12. A- K 5 202 LESSONS ON NUMBER. Answers to the Promiscuous Questions. Ans. 1. f. Am, 6. 17. 2. m. 7. 4i. 3. If 8. 4|f. 4. T ^. 9. 203 CHAPTER VII. PROPORTIONS AND PROGRESSIONS. 1. ARITHMETICAL PROPORTION, OR EQUI-D1FFERENCE. Teacher. What is the difference between two equal numbers? Pupils. Nothing. T. Name 2 numbers whose difference is 1. P. 1 and 2. T. Whose difference is 2? P. 1 and 3. T. Whose difference is 3 ? P. 1 and 4. T. Which of these is the greater ; the first or the second ? P. The second, namely, 4. T. Name 2 other numbers whose difference is also 3 ; and that the second be the greater of the two. P. 5 and 8. 204 LESSONS ON NUMBER. T. Hence, of the 4 numbers, 1, 4, 5, 8, it may be said that the difference between the 1st and 2nd is. the same as P. The difference between the 3rd and 4th. T. I will write this upon the slate. [Writing.] 1^4 = 5-8. Besides this, there is something else to be re- marked concerning these 4 numbers. What is it ? P. That the second is by as much greater than the 1st, as the 4th is greater than the 3rd. T. Can these 4 numbers be so placed, that the 1st is by as much greater than the 2nd, as the 3rd is greater than the 4th ? P. Yes, thus: 4-1 = 8-5. T. Find 4 other numbers of which the same may be said as of these 4. P. 2 - 5 = 7 - 10, or 5-2= 10- 7. T. What is the difference between the 1st and 2nd pair of these numbers ? P. 3. T. Hence it may be said, that their difference, 3, is common to each pair, or that they have a com- mon difference. Find 2 pair of numbers whose common differ- ence is 4. P. 6, 10, and 1 1, 15. T. How am I to place these 4 numbers, so that the order before mentioned may be observed ? PROPORTIONS AND PROGRESSIONS. 205 P. 6-10= 11-15, or 10- 6= 15- 11. 71 Can they be placed so that the difference be- tween each pair shall still be 4, and this order not be observed ? P. Yes; 6-10 = 15-11, or 10- 6 = 11-15. T. Now the 1st of these is less than the 2nd, but the 3rd is P. Greater than the 4th. T. Or, the 1st is greater than the 2nd, but the 3rd is P. Less than the 4th. T. Now, I wish you to find 2 pair of numbers whose common difference is 5, and to place them so as that the order before mentioned may be ob- served. P. 5-10= 11-16, or 10- 5= 16-11. T. Find 2 pair who^e common difference is 6. P. 7-13 = 15-21, or 13- 7 = 21-15. T. Two pair of numbers, similarly related, and placed in such an order, are said to form an arith- metical proportion, or equi-dijference. Now state how 2 pair of numbers must be re- lated, and how they must be placed, in order to form an arithmetical proportion. P. The difference between each pair must be 206 LESSONS ON NUMBER. the same ; and they must be so placed, that if the 1st is greater or less than the 2nd, the third is like- wise greater or less than the 4th. T. Find 6 arithmetical proportions. P. 7-3=9-5; 13-17 = 25-29; 18-12 = 20-26; 33-54= 17-38; 49 - 15 = 86 - 52 ; 100-80 = 30-10. T. Do 17 15 = 39 35 form an arithmetical proportion ? P. No ; because they have not a common differ- ence. T. Do 18 5 = 10 23 form an arithmetical proportion ? P. No; because, though they have a common difference, yet the 1st is greater than the 2nd ; but the 3rd is less than the 4th, which must not be. T. If, now, the 3 first numbers, or terms, as they are usually called, be known, you will be able, I think, to find the 4th term. For instance, the difference between 3 and 7 is the same as the dif- ference between 9 and what other number? P. 5 or 13. T. True ; but recollect that the four numbers must form an arithmetical proportion, and that I mentioned the lesser first. P. It must be 13, and not 5. PROPORTIONS AND PROGRESSIONS. 207 T. Hence 3-7=9-13; but had I said, the difference between 7 and 3 = the difference between 9 and what other number? your answer would have been P. 5. T. Let us try another question. [Writing on the slate.] 13-9 = 19- Is the 4th term to be less or greater than 19 ? P. Less than 19, because the 1st term, 13, is greater than the 2nd term, 9 ; hence the 3rd term, 19, must be greater than the 4th term, or the 4th term must be less than 19. T. And by how much must it be less than 19 ? P. By as much as 9 is less than 13 ; that is, by 4. T. How much, then, is the 4th term? P. 19-4, or 15. T. Hence 13-9 = 19-15. What, then, must be inquired into, in order to find the 4th term ? P. First, whether it is to be greater or less than the 3rd ; and then, by how much it is to be greater or less. T. Find a fourth number which shall form an equi-difference with 5, 11, 17. P. The difference between 5 and 11 = 6; and since 5 is less than 11, the 3rd term, 17, must be less 208 LESSONS ON NUMBER. than the 4th ; that is, it must be less than the 4th by 6 ; the number, therefore, is, 17 + 6 = 23. Hence, 5-11 = 17-23. T. Find a 4th number which shall form an equi- difference with the numbers -J-, |-, \. P. The difference between and since \ is greater than ^ by 1, therefore \ must be greater than the 4th term by -J-. Hence, \ = tV T% = A tne 4th term ; and therefore \ ^ = \ - ^. Similar solutions are required for each question ; and since the mind has to dwell some time upon each question, the teacher is advised to write the 3 numbers each time upon the school-slate, or let the pupils write the question upon their own slates. Answers to the Exercises. Arts. 4. 74. Ans. 10. 48. 5. 100. 11. T V 6. 33. 12. |f. 7. 51. 13. H. 8. 61. 14. 4 T V 9. 29. 15. 80 T V PROPORTIONS AND PROGRESSIONS. 209 2. ARITHMETICAL PROGRESSION. Teacher. Name 4 numbers whose successive dif- ferences are equal. Pupils. 1,2,3,4. T. What is the common difference ? P. 1. T. Name 4 other numbers whose successive dif- ferences are 2. P. 1, 3, 5, 7. T. What difference is there between these four numbers, and 4 numbers forming an equi-differ- ence? P. The numbers 1, 3, 5, 7 form an equi-differ- ence; but the difference, 2, is not only common to the 1st and 2nd pair, but also to the 3rd and 4th term. T. Name 4 other numbers whose successive dif- ferences are 3. P. 1, 4, 7, 10. T. Can there be more than 4 numbers having this property? P. Yes, as many as you please ; thus : 1, 4, 7, 10, 13, 16, 19, 22, &c. T. A series of numbers, whose successive differ- ences are equal, is called an arithmetical progres- sion. Form an arithmetical progression, the rium- 210 LESSONS ON NUMBER. bers, or terms, as they are called, having a common difference, 4. P. 1,5,9, 13, 17,21,25, &c. T. You made 1 the first term ; could the series begin with 2 ? P. Yes; 2,6,10, 14, 18, 22, &c. T. With 3 ? P. Yes; 3, 7, 11, 15, &c. T. In short, it might begin with any number. If, then, I tell you to form a progression, begin- ning with 5, and the difference of whose terms is 7, can you do it? P. Yes; 5, 12, 19, 26, 33, 8cc. T. If the 1st term of a progression be 2, and the 2nd term 7 ; what is the difference of the terms ? P. 5. T. And what is the progression? P. 2, 7, 12, 17, 22, 27, &c. T. The 1st term is 9 ; the 2nd term 16 ; what is the difference, and what the following terms ? P. The difference is 7, and the terms are, 9, 16, 23, 30, 37, 8cc. T. If the 1st term be 100 ; the 2nd, 97; what is the difference, and what are the terms of the progression ? P. The difference is 3, and the progression is 100, 97, 94, 91, 88, 85, 82, &c. T. What is there to be remarked concerning the PROPORTIONS AND PROGRESSIONS. 211 terms of this series, when compared with those of the former ? P. The terms in the series are decreasing ; in the former increasing. T. Hence, an arithmetical proportion may be P. Either increasing or decreasing. T. The 1st term of a series is 85; the 2nd 73. Is the series decreasing or increasing? P. Decreasing, because the 2nd term is less than the 1st. T. What are the terms of the series? P. The difference of the terms is 12 ; hence the terms are, 85, 73, 61, 49, 37, 8cc. T. If the 1st term be 1, and the common differ- ence be likewise 1, what is the series ? P. 1, 2, 3, 4, 5, See. T. What is the 10th term of this series? P. 10. T. If the 1st term be 1, the common difference 2, what is the 10th term? P. 19. T. How have you found this? P. We have written 10 terms of this progres- sion, thus : 1, 3, 5, 7, 9, 11, 13, 15, 17, 19. T. And, had I asked you to find the 100th term, what would you have done ? 212 LESSONS ON NUMBER. P. We should have drawn up 100 terms of the progression. T. This would have been rather troublesome ; now, if you will pay attention and reflect a little, we shall be able to find any term whatever, with- out first writing down all the previous terms. [Writing upon the slate :] 1. 2. 3. 4. 5. 6. 7. 1. 3. 5. 7. 9. 11. 13. 1. 4. 7. 10. 13. 16. 19. 2. 7. 12. 17. 22. 27. 34. What have I written here upon the slate? P. 7 terms of 4 progressions. T. The first of these may serve to show us the number of terms in the others; we will begin with the 2d series. Its first term is 1 ; what is the dif- ference of the terms ? P. 2. T. Now, tell me how the 2nd term 3 is ob- tained ? P. By adding the difference, 2, to the 1st term, 1. T. Is this true for each of the other two pro- gressions ? P. Yes ; for 1st term is 1, difference 3 ; and the 2d term is 1 +3 = 4; again, in the last progression, the 1st is 2, difference 5 ; and the second term is 2 + 5 = 7. PROPORTIONS AND PROGRESSIONS. 213 T. Hence the 2d term of every progression is equal to P. To the 1st term, more the difference. T. And how is the 3d term obtained from the 2d ? P. By adding the common difference 2 to it ; thus the 2d term is 3, difference 2 ; hence the 3d term is 3 + 2 = 5. T. Now, recollect, we ascertained before that the 2d term is the same as the 1st term, + the difference ; hence the 3d term must be P. The same as the 1st term, more the differ- ence; and more the difference again, that is, more twice the difference. T. Repeat what you have ascertained concern- ing the 2d and 3d term. P. The 2d term = ; 1st term +, the difference ; 3d term = ; 1st term +, twice the diff. T. See if this be true for each of the other 2 progressions. P. Yes, it is. T. Now observe how the 4th term is obtained from the 3d. P. By adding the common difference to it. T. And how is it obtained from the 1st term ? P. By adding 3 times the difference to the 1st term. T. Hence the 4th term is equal to 214 LESSONS ON NUMBER. P. The 4th term is equal to the 1st term, + 3 times the common difference. T. I think you will now be able to find out how the following terms are obtained from the 1st term. The pupils will thus find, that The 2d term = 1st term, + once the common difference. The 3d term = 1st term, + twice the common difference. The 4th term = 1st term, -I- 3 x the common difference. The 5th term = 1st term, + 4 x the common difference. The 6th term = 1st term, + 5 x the common difference. &c. The 10th term = 1st term, -f 9 x the common difference. &c. &c. If the series be increasing, but if decreasing, the difference must be subtracted from, instead of added, to the 1st term. T. If the 1st term be 1, and the common dif- ference 3, what will be the 10th term ? P. The 10th term= 1st term, +9x the dif- ference ; now the 1st term = 1, difference = 3 hence the 10th term = 1 +9 x 3 = 28. PROPORTIONS AND PROGRESSIONS. 215 T. By what means will you ascertain if this be true? P. By writing 10 terms of the progression, 1, 4, 7, &c. The pupils should be called upon to verify their answers for each question. Answers to the Exercises. Ans.3. 1; 7; 16; 19; 25; 31. 4. 100; 91; 82; 73; 64; 55. 5. J; f. 1_1_; 1|; !_?_ ; 2. 6. 20; 191- 5 18f ; 18; 17-J-; 16f. 7. See 2. Arithmetical Progression. 8. 43. Ans.12. 27f 9. 97. 13. 80. 10. 5i. 14. 293. 11. 6. 15. If. . GEOMETRICAL PROPORTION. Teacher. What does it mean to compare one thing with another ? Pupils. To ascertain in what respect the 2 things are alike, or in what respect they differ from each other. T. Hence, if we have to compare two numbers with each other, what would you do? 216 LESSONS ON NUMBER. P. Ascertain whether they are equal to each other, or not. T. Compare the numbers 2 and 6. P. They are not equal to each other. T. And if 2 numbers are not equal to each other, what may be said of them respectively? P. That the one is greater than the other; or that the one is less than the other. T. By what operation do you ascertain by how much 6 is greater than 2 ? P. By subtracting the lesser number 2 from the greater number 6. T. And what is the result of this operation called ? P. The difference between 6 and 2, which is 4. T. Is there any other way of comparing these two numbers with each other? P. Yes, by ascertaining how often the one is contained in, or contains the other. jT. By what operation do you ascertain how often 2 is contained in 6, or 6 in 2 ? P. By dividing 6 by 2, or 2 by 6. T. And what is the result of this operation called ? P. The quotient of 62, which is 3, or of 2-f-6, which is f or ^. T. Name 2 numbers, of which the difference is the same as that between 2 and 6. P. 12 and 16. PROPORTIONS AND PROGRESSIONS. 217 P. 12 and 16, T. Do you recollect what was said of 4 numbers similarly related as the numbers 2, 6, 12, 16? P. Yes ; they form an equi-difference, or arith- metical proportion. T. And can you find more pairs of numbers, whose difference is the same as that between 2 and 6? P. Yes, as many as you please; 14 and 18 ; 20 and 24, &c. T. Now, find 2 numbers, of which the quotient is the same as that of 6 divided by 2. P. 12 divided by 4; or 18-6 ; 30-r-10, &c. T. Hence what may be said of the 4 numbers 6, 2, 12, 4 ? P. That the 1st 6 divided by the 2d 2, is the same as the 3d 12 divided by the 4th 4. T. In short, that 6-f-2 = 124. Now find 2 numbers whose quotient is the same as that of l-f-2. P. 3-6, or 4-r-8, or 5--10, 8cc. T. How did you find these numbers ? P. Since l-r-2 = , any two numbers of which the one is -J- of the other must have a quotient = \ ; such numbers are 3-r-6, 48, 5-f-10. T. In short, then, l-r-2 = 3-i-6. Find 2 numbers whose quotient is the same as that of 2-r-l. P. Because 2-rl =2; any two numbers, of which the one is twice as much as the other, must L 218 LESSONS ON NUMBER. have a quotient = 2 ; such are 6-r-2, 8-r-4, lO-f-5, 8cc. T. We have found before that l-f-2 = 3-r-6 ; and now, that 2-r-l = 6-r-3. What conclusion can you make ? P. That of 4 numbers, if the 1st divided by the 2nd, is equal to the 3rd divided by the 4th, then also is the 2nd divided by the 1st, equal to the 4th divided by the 3rd. T. Any 4 numbers having this property are said to be proportional to each other. So that it may be said, 1 has the same proportion to 2, as 3 has . . . . to 6, or, as 4 has . . . . to 8. &c. And that, 2 has the same proportion to 1, as 6 has . . . . to 3, or, as 8 has . . to 4. &c. And this is shortly written thus : 1 : 2 : : 3 : 6. 1 : 2 : : 4 : 8. 2 : 1 : : 6 : 3. And 2 : 1 : : 8 : 4. PROPORTIONS AND PROGRESSIONS. 219 Which is read thus : as 1 is to 2, so is 3 to 6 ; as 2 is to 1, so is 6 to 3. &c. Now, find 2 numbers which have the same pro- portion that 1 has to 3. P. 2 to 6; 3 to 9; 4 to 12, &e.; or any num- bers of which the first is ^ of the other. T. How have you ascertained this ? P. By dividing 1 by 3, which is |-, and any two numbers of which the first is % of the other, are in the proportion of 1 to 3. T. And, if you have to find two numbers in the proportion of 3 to 1, which are they ? P. 6 to 2 ; 9 to 3 ; 12 to 4, &c. ; or any two numbers of which the first is 3 times as much as the other. T. Hence, to ascertain two numbers, which have the same proportion to each other as two other numbers, what must be done? P. Divide the one by the other ; and find two numbers, which, when divided, have the same quotient. jT. Speaking of proportional numbers, this quo- tient is called the ratio, which the two numbers have to each other. What is the ratio of 6 to 12 ? P. A or i- T. Arid what is the ratio of 12 to 6 ? P. V 2 or 2. L 2 220 LESSONS ON NUMBEK. T. And of 4 numbers which form a proportion, if the 1st be greater or less than the 2nd, the 3rd must be P. Greater, or less than the 4th. T. Hence, as 1 : 4 : : 5 to what other number? Is it to be less or greater than 5? P. It must be greater than 5, since the first 1 is greater than 4, the 2nd. T. What is the number? P, 20 ; since the ratio of 1 to 4 is J-, hence the number of which 5, is i, is the number required ; which, therefore, is 4x5, or 20. ' T. This is called finding the 4th proportional to the 3 numbers 1, 4, 5. Find the 4th proportional to 4, 1, 5. P. Since 4 = 4, the ratio of 5 to the number must be 4, that is, 5 must be 4 times as much as the number, which is, therefore, - of 5, or 1^ ; hence, 4 : 1 : : 5 : 1-J-. T. As 2 is to 3, so is 4 to what 4th number? P. Because 2 is f of 3, 4 must be f of the number ; hence -J- of the number = -J- of 4 ; and therefore the number must be \ of 4 taken 3 times, which is 6 ; therefore, 2 : 3 : : 4 : 6. N.B. The pupils will not find it difficult to ascertain the 4th proportional, when the 1st term is either a multiple of the 2nd, or the 2nd a mul- tiple of the first. When, however, the ratio of the PROPORTIONS AND PROGRESSIONS. 221 first two terms is f, f , f , f , f , , &c. that is, fractional, they will naturally experience greater difficulty. Before entering then upon similar questions, it will be advisable to go through the following exercises. 6 is the half of what number? Ans. Of 2 x 6 ; that is, of 12. 6 is the third part of what number ? Ans. Of 3x6; that is, of 18. 6 is f of what number? Ans. Of the half of 6 taken 3 times; that is, of 9. 6 is i of what number ? Ans. Of 4 x 6 ; that is, of 24. 6 is of what number ? Ans. Of ^ of 6 taken 4 times ; that is, of 8. 6 is ^ of what number ? Ans. Of 5x6; that is, of 30. 6 is f of what number? Ans. Of i of 6 taken 5 times; that is, of 15. 6 is % of what number? Ans. Of % of 6 taken 5 times ; that is, of 10. 6 is ^ of what number? Ans. Of ^ of 6 taken 5 times; that is, f x 5 = = 7-1, L 3 222 LESSONS ON NUMBER. 6 is 4 of what number ? Am. Of ^ of 6 taken 7 times ; that is, 11 is |^ of what number ? Ans. Of ^ of 11 taken 8 times; that is, y x 8 =' \s = 29|. &c. &c. These and similar questions having been duly practised, no difficulty will be met with in the following : Teacher. As 7 is to 1, so is 13 to - Pupils. l. Because the 1st term is 7 times as much as the 2nd, the 3rd must be 7 times as much as the 4th. Now 13 is 7 times y, which is If T. As 7 is to 2, so is 8 to - P. 2f because 7 is of 2, therefore 8 must be \ of the number. Now 8 is \ of \ of 8 taken twice ; that is, f x 2, or V 6 = 2f . T. If your answer, 2f, be correct, what must be- P. 7, divided by 2, must give the same quotient as 8, divided by 2-f : 7-2 = -I, and 8-7- 2f = 8~ V = i- Hence the answer is correct. PROPORTIONS AND PROGRESSIONS. 223 Answers to the Exercises. N.B. The answers to the 7 first questions are contained in 3. Geometrical Proportions. Am. 8. Ratio of 3 to 7=4. Ratio of to 7 to 3 = = f . T' 8 to 11 = A- f tof = 11 to 8 = V- 1 tof = ito * = i ~5 tof = ,!HS. 9. ] 5 7 35. 10. 5 1 9 1 I- 11. 4 12 13 39. 12. 12 4 13 4.1. 13. 15 60 3 12. 14. 9 72 14 112. 15. 72 9 14 1|. 16. 17 51 23 69. 17. 18 90 85 425. 18. 19 114 105 630. 19. Of 1 5. 20. Of 1 6. 21. Of 1 Of. 22. Of 8 i- 23. Of 2 1. 24. Of 3 25. Of 2 0. 224 LESSONS ON NUMBER. Am. 26. 2 3 9 13$. 27. Q 2 9 6. 28. o 4 15 20. 29. 4 3 15 "i- 30. 4 5 18 22$. 31. 5 4 18 14|. 32. 6 7 5 55. 33. * 5 * 3*. 34. I f 6 Printed by Stewart and Co., O/d Bailey. PRINTED FOR JOHN TAYLOR, Boofcseller autr f ufclisfjer to tfje $%iubersttg of Eonirott, UPPER GOWER STREET. LESSONS ON NUMBER, As given at a Pestalozzian School at Cheam, Surrey. Second Edition. Part II. THE SCHOLAR'S PRAXIS. ft. THE ELEMENTS OF ARITHMETIC. ByAuGUSTUsDE MORGAN, Esq. Second Edition, considerably enlarged, 12mo. 3*. Gd. cloth. " Instead of merely learning a number of rules by rote, the pupil learns to refer every thing to reason, and he is taught how to do so ; and thus he will often be able to ascertain the meaning of an am- biguous passage, or supply the defect of an imperfect proof. It is only thus that any knowledge of the principles of Arith- metic can be acquired ; and although it is not necessary to resort to these consider- ations to attain practical facility in per- forming arithmetical operations, we have little doubt that this will be gained quite as rapidly under instruction, such as Pro- fessor De Morgan's Treatise affords, as it is from the old course of mere rules and examples, and it will certainly be gained better." Journal of Education, No. I. " Since the publication of the first edition of this work, though its sale has suffici- ently convinced me that there exists a disposition to introduce the principles of arithmetic into schools, as well as the practice, I have often heard it remarked that it was a hard book for children. I never dared to suppose it would be other- wise. All who have been engaged in the education of youth are aware that it is a hard thing to make them think ; so hard, indeed, that masters had, within the last few years, almost universally abandoned the attempt, and taught them rules instead of principles ; by authority, instead of demonstration. This system is now pass- ing away ; and many preceptors may be found who are of opinion that, whatever may be the additional trouble to them- selves, their pupils should always be in- duced to reflect upon, and know the reason of, what they are doing. Such I would advise not to be discouraged by the failure of a first attempt to make the learner un- derstand the principle of a rule. It is no exaggeration to say, that, under the pre- sent system, five years of a boy's life are partially spent in merely learning the rules contained in this treatise, and those, for the most part, in so imperfect a way, that he is not fit to encounter any ques- tion unless he sees the head of the book under which it falls. On a very moderate computation of the time thus bestowed, the pupil would be in no respect worse off, though he spent five hours on every page of this work. The method of proceeding which I should recommend, would be as follows: Let the pupils be taught in classes, the master explaining the article as it stands in the Work. Let the former, then, try the demonstration on some other numbers proposed by the master, which should be as simple as possible. The very words of the book may be used, the figures being changed, and it will rarely be found that the learner is capable of making the proper alterations, without understanding the reason. The experience of the master will suggest to him various methods of trying this point. When the principle has been thus discussed, let the rule be dis- tinctly stated by the master, or some of the more intelligent of the pupils ; and lot some very simple example be worked at length. The pupils may then be dismissed, to try the more complicated exercises with which the work will furnish them, or any others which may be proposed." Preface- Mathematical Works published by John Taylor. BARLEY'S SCIENTIFIC LIBRARY, FOR THE USE OF SCHOOLS, PRIVATE STUDENTS, ARTISTS, AND MECHANICS. It is the purpose of this Work to furnish a Series of Elementary Trea- tises on Mathematical Science, adapted to the wants of the public at large. To youth of either sex at public and private schools ; to Persons whose education has been neglected, or whose attention has not been directed in early life to such studies ; and to Artists and Mechanics, these little works will be found particularly suited. The principles of the various Sciences are rendered as familiar, and brought as near to our commonest ideas as possible ; the demonstrations of propositions are made plain for the mind, and brief for the memory; and the Elements of each Science are reduced not only to their simplest but to their shortest form. XIII. A SYSTEM OF POPULAR GEOMETRY. Containing in a few Lessons so much of the Elements of Euclid as is necessary and sufficient for a right understanding of every Art and Science in its leading Truths and general Principles. By GEORGE DARLEY,A.B. Third Edition. Price 4*. 6d. cloth. XIV. COMPANION TO THE POPULAR GEOMETRY. In which the Elements of Abstract Science are familiarised, illustrated, and rendered practically useful to the various purposes of Life, with numerous Cuts. Price 4*. Qd. cloth. xv. A SYSTEM OF POPULAR ALGEBRA, With a Section on Proportions and Progressions. Second Edition. Price 4*. Q