fourth of what number?" "What is 2-thirds of 12?" "4-fifths of 25 are how many times 6?" These operations are, in general, introduced by practical examples. Abstract examples are then given for drill, and more practical examples for application. At the end 59 miscellaneous problems are given.

The last section of the text is devoted to the tables of Federal money, dry measure, wine measure, Troy weight, apothecaries' weight, avoirdupois weight, long measure, cloth measure, square measure, measure of time, and sterling money. The plan of treatment is first the table and then questions for drill. Most of these are in the form, "How many quarts in 1 peck? 2 3 a few of the questions approach practical problems.

2

4?" Only

In 1843 the Little Arithmetic was revised and enlarged and was published under the title "Ray's Arithmetic, Part Second." The first 54 pages are identical with the first 57 pages of the Little Arithmetic except that the first problems in addition, subtraction, multiplication, and division are illustrated by small circles. Beginning on page 55, fractions are presented again more formally, but the author still retains much of the form and spirit of the earlier pages. The fractions are represented by dividing a "yard of tape." Pages 97 to 144 are given to "Practical Arithmetic." This includes notation and numeration up to nine places, the four operations for integers, reduction, and the four operations for denominate numbers, simple proportion, or the rule of three, and simple interest. In general, the presentation is formal, a single practical problem followed by the definition, explanation of the solution, statement of the rule, and abstract exercises for drill. Practical problems as applications are placed last.

In 1857 Ray's Arithmetic, Part Second, had the title, "Intellectual Arithmetic by Induction and Analysis." The important changes are "appropriate models of analysis and frequent reviews,” the introduction of percentage, gain and loss, interest, and their applications, and the addition of a number of difficult problems. The "appropriate models of analysis" are given following the first problem of a lesson and again when a new type of problem is encountered. There is a tendency to place the abstract work before the practical problems. This is particularly true in the topics which have been added.

The edition of 1877 contains few significant changes. Objective illustrations are omitted. The presentation of fractions is more formal, the definition being given first and the logical order is approached. The space given to percentage and its applications is increased.

1 Compare these exercises with the ones used by Pestalozzi. See p. 59. Ray followed Pestalozzi even more closely than did Colburn.

• These pages include all of the text except the section devoted to denominate numbers.

The content of the mental arithmetics followed very closely that of Colburn's First Lessons. The four operations for integers and for vulgar fractions, a few of the most important tables of denominate numbers, percentage, and interest would serve well as a table of contents for any mental arithmetic. The only change necessary would be in the order and emphasis.

The topics, as in the case of Ray's text, are presented in very much the same fashion as in the First Lessons. Each is introduced by practical problems, which are followed by abstract ones for drill. This order is almost invariably retained, even in revised editions and texts published well toward the close of this period. The plan of following a practical problem by the same combination with abstract numbers was not followed except during the active period, and the number of abstract drill exercises were relatively less during the static period. Much space was given to review questions, miscellaneous problems, and promiscuous examples. This was evident attempt to secure thoroughness.

an

The practical problems are essentially of the same quality as those in Colburn's text. In fact, those of some of the texts bear a very close resemblance to those of the First Lessons. Toward the close of the period there is a noticeable increase in the difficulty of the problems. They were made more difficult in two ways: First, the magnitude of the quantities was made greater; second, the problems themselves were made intricate. The extent to which this was carried is shown in the following problems selected from Ray's Intellectual Arithmetic, one thousandth edition, 1860:

A hare is 100 leaps before a hound and takes 5 leaps while the hound takes 3, but 3 leaps of the hound equal 10 of the hare; how many leaps must the hound take to catch the hare?

A trout's head is 4 in. long, its tail is as long as its head and of its body, the body is as long as its head and tail; what is its length?

If 10 gal. of water per hr. run into a vessel containing 15 gal. and 17 gal. run out in 2 hr., how long will the vessel be in filling?

A, B, and C rent a pasture for $92. A puts in 4 horses for 2 mon., B 9 cows for 3 mon., and C 20 sheep for 5 mon. What should each pay, if 2 horses eat as much as 3 cows and 3 cows eat as much as 10 sheep?

If the interest for 1 year 4 mon. is 3/25 of the principal, what is the interest of $100 for 1 yr., 8 mon., 18 da.?

The number and difficulty of such problems varied with the author. Ray probably represents an average, certainly not less than an average. Texts containing difficult problems seem to have been demanded, particularly in the latter part of this period. In his New Mental Arithmetic, 1873, Brooks gives a large number of problems which are nothing more than intricate puzzles. He classifies them under such heads as pasture problems, beggar and equal number problems, animal problems, working problems, labor and fish problems, age and step problems, etc.

- 81758°-17—8

Practical arithmetic. The texts which are grouped under this head included all the topics of arithmetic which were studied in the elementary school. They began with numeration and notation, and addition, subtraction, etc., came in turn. In scope they were the descendants of such texts as those of Dilworth, Daboll, and Adams. The study of "practical arithmetic" paralleled that of "mental arithmetic."

Colburn applied his ideas of arithmetic, particularly the inductive method, to this field and, as we have shown, produced a text of high merit. But the Sequel was not well received, and after a few years dropped out of notice. The book embodied some features, such as the entire departure from the traditional division of subject matter and the order of topics, which were too progressive for the times. Furthermore, the book had to compete with contemporary texts and texts which were already in use. This was not true of the First Lessons, for it was a pioneer in a new field. Thus Colburn probably influenced only very slightly this field of arithmetic directly through the Sequel. However, the work of Colburn did change the texts in practical arithmetic. The source of this influence was primarily the First Lessons. The principles underlying this text were accepted, and other writers, like Colburn, attempted to apply them in part to the more advanced texts.

We have described the Scholar's Arithmetic by Daniel Adams. In 1827, he says in the preface of Adams's New Arithmetic:

The Scholar's Arithmetic, published in 1801, is synthetic. If that is a fault of the work, it is a fault of the times in which it appeared. The analytic or inductive method of teaching, as now applied to elementary instruction, is among the improvements of later years. Its introduction is ascribed to Pestalozzi, a distinguished teacher in Switzerland. It has been applied to arithmetic, with great ingenuity, by Mr. Colburn, in our country.

The analytic is unquestionably the best method of acquiring knowledge; the synthetic is the best method of recapitulating, or reviewing it. In a treatise designed for school education both methods are useful. Such is the plan of the present undertaking, which the author, occupied as he is with other objects and pursuits, would willingly have forborne, but that, the demand for the Scholar's Arithmetic still continuing, an obligation, incurred by long-continued and extended patronage, did not allow him to decline the labor of the revisal, which should adapt it to the present more enlightened views of teaching this science in our schools. In doing this, however, it has been necessary to make a new work.

Division is introduced with the problem, "James divided 12 apples among 4 boys; how many did he give each boy?" After 20 problems of this sort, the problem, "How many oranges, at 3 cents each, may be bought for 12 cents?" is solved by successive subtractions. The pupil is then told, "We may come to the same result by a process, in most cases much shorter, called Division." The process of division is explained by solving this problem, and the new words are defined. The division table is given, and after the

problem, "How many yards of cloth, at 4 dollars a yard, can be bought for 856 dollars?" is solved and explained, the rule is stated for the case when the divisor does not exceed 12. The rule for the case when the divisor exceeds 12 is derived in the same manner.

A comparison of this presentation of division with the way in which such topics were presented in the texts of the preceding period indicates the extent of Colburn's influence upon the the "practical arithmetics." The inductive method was accepted with only slight reservation, and Adams seems to have caught something of the spirit of it, as well as the form.

Roswell C. Smith, whose arithmetic was first published in 1827, says in a rather bombastic preface in the third edition, 1834:

Another inquiry may still be made: Is this edition different from the preceding? The answer is, Yes, in many respects. The present edition professes to be strictly on the Pestalozzian, or inductive, plan of teaching. This, however, is not claimed as a novelty. In this respect, it resembles many other systems. The novelty of this work will be found to consist in adhering more closely to the true spirit of the Pestalozzian plan; consequently, in differing from other systems, it differs less from Pestalozzian. This similarity will now be shown.

The author attempts to combine oral and written arithmetic. Certain features of the oral part of the text almost duplicate Colburn's First Lessons both in actual content and spirit. For example, Smith begins his text with:

1. How many little fingers have you on your right hand? How many on your left? How many on both?

2. How many eyes have you?

3. If you have two apples in one hand, and one in the other, how many have you in both? How many are two and one, then, put together?

4. How many do your ears and eyes make, counted together?

5. If you have two nuts in one hand, and two in the other, how many have both? How many do two and two make, put together?

you in

The Hindu numerals are introduced on page 2 and the addition tables are given on pages 3 and 4. Aside from a list of 24 problems, there is no work preceding the tables, and these problems do not constitute a development of the addition facts. Following the table, there is only a page of drill. The remaining three operations are disposed of in the same manner. This completes the "mental exercises." Beginning on page 17 we find the traditional order of topics, numeration and notation, addition, etc. The fundamental operations are introduced by a list of practical problems to be solved mentally. The "interrogative system" is used throughout the work in presenting rules and explanation. The author did this under the impression that it was the mark of inductive presentation, but nevertheless the subject is presented quite dogmatically. For the most part the spirit of the book is deductive rather than inductive.

1

1 This same interrogative, or question and answer, system was used in Dilworth's Schoolmaster's Assistant. Therefore this feature is not necessarily due to Pestalozzi.

In presenting a new process, e. g., long division, one problem is solved and explained, after which the rule is stated.

The text, while it possesses some merit and must be considered one of the progressive texts of its time, does not reflect much of Pestalozzian principles and does not equal Colburn's texts in this respect.

The plan of the written arithmetic of Emerson's North American Arithmetic, Part Second, is much like the oral part. But the structure of the book is more formal, objective illustration is lessened, and the inductive method, although retained, has lost much of its spirit. The point of departure in taking up a new process is not always a concrete problem, and the development is forced. For example, in division:

How many yards of cloth, at 3 dollars a yard, can be bought for 396 dollars? Here we must find how many time 3 dollars there are in 396 dollars; that is, we must 3)396 divide 396 by 3. 132

We first divide the 3 hundreds, then the 9 tens, and then

the 6 units; thus, 3 in 3, once; 3 in 9, 3 times; 3 in 6, 2 times.

Observe, in the above example, that the 3 which we first divide means 3 hundred; and the 1 which we place under it means 1 hundred, showing that 3 is contained in 300, 100 times. The 9 means 9 tens, and the 3 which we place under it means 3 tens, showing that 3 is contained in 90, 30 times.

A dividend is a number which is to be divided; such as the number 396 in the above example. A divisor is a number by which we divide; such as the number 3 in the above example. The quotient is the number of times which the divişor is contained in the dividend; such as the number 132 in the above example.

Long division comes four pages later. The topic is introduced as follows:

The method of dividing taught in the two preceding sections is called short division; the method taught in this section is called long division. In long division we place the quotient on the right hand of the dividend, and perform the same operations under the dividend, heretofore performed in the mind.

4)95307(23826 How many times is 4 contained in 95307?

8

15

12

33

32

10

8

27

Perceiving that 4 is contained in 9 twice, we place 2 in the quotient, multiply the divisor by 2, and subtract the product (8) from 9. This is the same as saying in short division, “4 in 9, 2 times and 1 over.” Now, since the 1 over must be joined with the 5, we bring down the 5 to the right of the 1; and then, perceiving that 4 is contained in 15, 3 times, we place 3 in the quotient, multiply the divisor by 3, and subtract the product as before. Thus we proceed to bring down every figure of the dividend and unite it with the previous remainder.

24

3

Additional difficulties encountered in division are explained in the same manner, which is essentially only a detailed rule stated for a particular example. After 24 examples, all abstract, the general