together with some complications of the preceding combinations. It is suggested that this section may be omitted until reviewing the book. The next four sections contain questions which call for special cases of the division of an integer or a mixed number by a fraction, as, "How many one-thirds in 4?" or "How many thirds in two and one-third?" and for the multiplication of fractions and mixed numbers by integers in general.

In Section XII the symbolism of fractions is given for the first time, and exercises upon the combinations of the preceding four sections are stated in terms of fractional symbolism. In the following sections these operations are taken up in the order named: Reduction of fractions to a common denominator, addition and subtraction of fractions, reduction of fractions to lowest terms, division of fractions by whole numbers, multiplication of one fraction by another, division of whole numbers by fractions, and division of one fraction by another.

Colburn remarks that the division of a fraction by a whole number calls for the same operation as the multiplication of a fraction by a fraction. For this reason he places together the problems which require these combinations. He also points out that it is difficult to find problems which require a fraction to be reduced to its lowest terms. For this operation he gives only abstract examples, but he suggests that it would be well to omit this article the first time the pupil goes through the book, and "after he has seen the use of the operation let him study it."

The tables of Federal money, sterling money, troy weight, avoirdupois weight, cloth measure, wine measure, dry measure, the measure of time, and a list of 183 miscellaneous problems are given as a sort of an appendix.

In such problems as the following the notion of rate is expressed:

A man failing in trade was able to pay his creditors only 4 shillings on a dollar; how much would he pay on 2 dollars? How much on 3 dollars? How much on 7 dollars? How much on 10 dollars?

Interest is introduced with a note which explains the meaning of the term. After explaining that "6 per cent" means 6 cents on a dollar, 6 dollars on a hundred dollars, or 6 pounds on a hundred pounds, he makes the generalization that it is "6/100 of the sum, whatever the denomination." The pupil is given such problems as:

The interest of 1 dollar being 6 cents for 1 year, what is the interest of 7 dollars for the same time? What is the interest of 10 dollars? Of 15 dollars? Of 20 dollars? Of 30 dollars? Of 50 dollars? Of 75 dollars? Of 100 dollars? Of 118 dollars?

Finally, the pupil is given such as these to solve:

If the interest of 2 months or 60 days is 1 per cent, what would be the per cent for 20 days? What for 40 days? What for 15 days? What for 45 days? What for 12 days? What for 10 days? What for 5 days?

What is the interest of 100 and 37 dollars for 2 years 3 months and 20 days?

Fellowship is presented by such problems as:

Two men bought a bushel of corn, one gave 1 shilling, the other 2 shillings; what part of the whole did each pay? What part of the corn must each have?

Two men hired a pasture for 58 dollars; one put in 7 horses, and the other 3 horses;

what ought each to pay?

Three men commenced to trade together; they put in money in the following proportion; the first, 3 dollars, as often as the second put in 4, and as often as the third put in 5; they gained 87 dollars. What was each man's share of the gain?

Two men hired a pasture for 32 dollars. The first put in 3 sheep for four months, the second put in 4 sheep for five months; how much ought each to pay?

Following this last problem, which is the first in double, or compound fellowship, an explanatory note of five lines is given. In the case of simple fellowship no explanation is given.

There are a few arithmetical puzzles of which the following are typical:

Said Harry to Dick, my purse and money together are worth 16 dollars, but the money is worth 7 times as much as the purse; how much money was there in the purse? and what is the value of the purse?

A man having a horse, a cow, and a sheep, was asked what was the value of each. He answered that the cow was worth twice as much as the sheep, and the horse 3 times as much as the sheep, and that all together were worth 60 dollars. What was the value of each?

A man driving his geese to market was met by another, who said, "Good morrow, master, with your hundred geese." Says he, "I have not a hundred, but if I had half as many more as I now have, and two geese and a half, I should have a hundred." How many had he?

1

What number is that, to which if its half and its third be added the sum will be 55? Objective materials. In the Key directions are given for using the Pestalozzian tables and other objective materials. Before 1821, children used their fingers, and even their toes, in learning to count, and probably counted out problems on them. But this practice seems to have been tolerated rather than recognized as a legitimate and valuable method of learning number facts. Certain it is that Colburn was the first author in the United States to introduce objective materials in an arithmetic text. The plates represent just one type of objects which he used. Beans, grains of corn, pieces of crayon, marks, etc., are recommended for use and even preferred. He says:

The first examples may be solved by means of beans, peas, etc., or by Plate I. The former method is preferable, if the pupil be very young, not only for the examples in the first part of this section, but for the first examples in all the sections.2

Mental arithmetic.-Colburn's First Lessons is an "intellectual" arithmetic, i. e., the examples are to be solved without the use of pencil and slate or paper. The Hindu symbols for writing numbers are not given until page 50, and methods of calculating with figures are not given. Numbers greater than 100 occur in very few problems, but within this quantitative range Colburn has treated many

1 See p. 57 for a description of these tables.

* Key to First Lessons, p. 144.

of the topics which we have found in texts of the ciphering-book period. A comparison reveals the following: Notation, the four operations for integers, practically all of the important denominate numbers and the operations upon them, common fractions completely, rule of three, direct and inverse, barter, practice, single and double fellowship, and interest. The important omissions are decimal fractions, exchange, evolution, loss and gain, and alligation. The topics omitted have to do with situations with which young children are relatively unacquainted. Exchange, and loss and gain dealt with situations peculiar to a professional business man. Decimal fractions were tools of calculations of business or of evolution. Alligation was an obsolescent topic.

Summary.-Colburn says of the plan of this book that it "entirely supersedes the necessity of any rules, and the book contains none.' The child is to be given a practical example and from his understanding of the situation involved he is to decide upon the operation or operations to be performed. If he can not do this when the numbers are made simple, Colburn says that he is not ready for such an example. Colburn held that abstract exercises were more suitable for reducing the combinations to the level of habit than practical examples. And it is this function which the abstract examples were intended to fulfill.

Upon completing Colburn's First Lessons, a pupil was acquainted with a large per cent of the quantitative situations which he would probably meet in life. He had met practical examples taken from these situations, and he had had to decide upon the combinations to be made. In this way he came to understand the situations so that he knew what combinations should be made, even though the quantities should be so large as to require written calculations. He had learned as much of notation and the symbolisms of arithmetic as he has needed. He knew the denominate quantities which he had met in the practical examples. And he had been thoroughly drilled upon the fundamental number facts.

THE SEQUEL.

As its title indicates, the Sequel was intended to be studied by the pupil after he had completed the First Lessons. Colburn states in the preface to the Sequel that the pupil may commence the First Lessons as soon as he can read the examples or perhaps even before. By doing this the pupil would be prepared to commence the Sequel by the time he was 8 or 9. It was written to be a practical arithmetic, but Colburn expected the pupil to learn something of the science of arithmetic as he worked with practical examples.

In his analysis of the subject matter of arithmetic, Colburn distinguished between the processes of arithmetic, which he calls "prin

ciples," and the application of arithmetic, which he designates as 'subjects." To him the "principles" mean arithmetic and the applications merely a field for the exercise of these principles; denominate numbers, mensuration, percentage, interest, etc., are not taken. as the basis for separate chapters, or even distinct topics. "To give the learner a knowledge of the principles" is his purpose, and to this end the problems are grouped about the principles.

Colburn takes the position that "When the principles are well understood, very few subjects will require a particular rule, and if the pupil is properly introduced to them, he will understand them better without a rule than with one." For example, if a pupil understands well the relation between a product and its factors in all its phases, percentage and its applications require no particular rule and will present no difficulty to the learner. At most the learner will need to be told the meaning of the new terms used in expressing the problem. As would naturally be expected from such a point of view, the applications of arithmetic do not influence nor determine the organization of Colburn's texts.

The plan of the Sequel.-The subdivisions and order of the "principles" are unusual. Multiplication of integers follows addition instead of subtraction. In fractions, multiplication is placed first and is followed by addition and subtraction. Both multiplication and division of fractions are divided into several cases. The Sequel is divided into two parts. The first consists of graded lists of problems with an occasional suggestive note to define some new term or to interpret the meaning of the problem. "The second part contains a development of the principles" based upon problems.

The two parts are to be studied together, when the pupil is old enough to comprehend the second part by reading it himself. When he has performed all the examples in an article in the first part, he should be required to recite the corresponding article in the second part, not verbatim, but to give a good account of the reasoning. When the principle is well understood, the rules which are printed in italics should be committed to memory.

Colburn gives rules only for the principles and not for the applications of arithmetic. The table of contents of the Sequel makes no mention of any of the applications of arithmetic, several of which usually have a chapter devoted to them.

Colburn mentions the following "subjects" as being specifically included in the text: Compound multiplication, addition, subtraction, and division; simple interest, commission, insurance, duties and premiums, common discount, compound interest, discount, barter, loss and gain, simple fellowship, compound fellowship, equation of payments, alligation medial, alligation alternate, square and cubic

1 Few rules are given, and such as are given are placed at the end of a section. It is intended that the pupil will develop the rule as the result of solving problems before he reaches the printed statement.

measure, duodecimals, taxes, mensuration, geographical and astronomical questions, exchange, tables of denominate numbers.

Topics omitted. Colburn omits some topics entirely. He specifically mentions the rule of three, position, and powers, and roots. The reasons he gives for their omission are:

Those who understand the principles sufficiently to comprehend the nature of the rule of three, can do much better without it than with it, for when used, it obscures rather than illustrates the subject to which it is applied.

*** This (rule of position) is an artificial rule, the principle of which can not be well understood without the aid of algebra; and when algebra is understood, position is useless. Besides, all the examples which can be performed by position may be performed much more easily, and in a manner perfectly intelligible without it. Powers and roots, though arithmetical operations, come more properly within the province of algebra.

It is interesting to note that some of the omissions which Colburn made nearly a century ago are still considered debatable by some teachers.

How the "principles" are presented.-A masterly exposition of our decimal system of numeration is given in which Colburn shows its function. After defining the numbers 1 to 10 as names given to collections of units, he continues:

In this manner we might continue to add units, and to give a name to each different collection. But it is easy to perceive that if it were continued to a great extent it would be absolutely impossible to remember the different names; and it would also be impossible to perform operations on large numbers. Besides, we must necessarily stop somewhere; and at whatever number we stop, it would still be possible to add more; and should we ever have occasion to do so, we should be obliged to invent new names for them, and to explain them to others. To avoid these inconveniences, a method has been contrived to express all the numbers that are necessary to be used, with very few names.

The first ten numbers have each a distinct name. The collection of ten simple units is then considered a unit; it is called a unit of the second order. We speak of the collections of ten, in the same manner that we speak of simple units; thus we say one ten, two tens, three tens, four tens, five tens, six tens, seven tens, eight tens, nine tens. These expressions are usually contracted; and instead of them we say ten, twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety.

To express the numbers between the tens, the numbers below ten are to be added to the tens. Colburn then explains how the names of numbers which are used in common language have been derived by such a method. After telling how a hundred and a thousand are made up he indicates how "this principle may be continued to any extent," and expresses his admiration of the decimal system of numeration by saying:

Hence it appears that a very few names serve to express all the different numbers which we ever have occasion to use. To express all the numbers from one to nine thousand, nine hundred and ninety-nine, requires, properly speaking, but "twelve" different names.' It will be shown hereafter that these twelve names express the numbers a great deal further.