The "Arabic" and Roman methods of writing numbers are carefully explained in 11 pages. The Roman system is given in a footnote, with the statement that "a short description of it may be interesting to some." In Part I it is not mentioned.

Although it was Colburn's plan that the pupil should study the First Lessons before commencing the Sequel, yet he wrote the Sequel in such a way that this would not be "absolutely necessary." For example, in the development of addition he begins with a problem any child who is old enough to study the book can understand. After defining addition as putting together two or more numbers to ascertain what numbers they will form," he gives the problem: "A person bought an orange for 5 cents and a pear for 3 cents; how many cents did he pay for both?" This problem is solved by taking the 5 and joining the 3 "to it a single unit at a time."

Says Colburn:

A child is obliged to go through the process of adding units every time he has occasion to put two numbers together until he can remember the results. This, however, he soon learns to do if he has frequent occasions to put numbers together.

He also points out that the child can not make much progress in arithmetic until he learns perfectly the addition tables up to ten.

Colburn's development of carrying in addition is based upon the decimal structure of the system of numeration. The first practical example calls for 24 and 8 to be added. He points out that 24 is simply an abbreviation for 20 and 4. "To add eight to twenty-four, add eight and four, which are twelve. To twelve add twenty. But twelve is the same as ten and two, therefore we may say twenty and ten are thirty and two are thirty-two." Three more practical examples, each one becoming more difficult, are explained in the same way. He then defines "carrying" by saying: "The reserving of the tens, hundreds, etc., and adding them with other tens, hundreds, etc., is called carrying." The principle of carrying is further illustrated by the following example, whose solution he explains:

A merchant had all his money in bills of the following description, one-dollar bills, ten-dollar bills, hundred-dollar bills, thousand-dollar bills, etc.; each kind he kept in a separate box. Another merchant presented three notes for payment, one 2,673 dollars, another 849 dollars, and another 756 dollars. How much was the amount of all the notes; and how many bills of each sort did he pay, supposing he paid it with the least possible number of bills?

Additional illustration of the principle of carrying is given by writing the addends in this form: 4000+ 600+ 70+ 3. And finally when he is ready to state the rule, Colburn says: "From what has been said, it appears that the operation of addition may be reduced to the following rule."

Multiplication immediately follows addition and is begun with this example: "How much will 4 gallons of molasses come to at 34

cents a gallon?" After the example is solved by addition, it is pointed out that "if it were required to find the price of 20, 30, or 100 gallons, the operation would become laborious." Colburn goes on to say:

If I have learned that 4 times 4 are 16, and that 4 times 3 are 12, it is plain that I need not write the number 34 but once, and then I may say 4 times 4 are 16, reserving the 10 and writing the 6 units as in addition. Then again, 4 times 3 (tens) are 12 (tens) and 1 (ten which I reserved) are 13 (tens).

Multiplication is then defined as "addition performed in this manner."

Subtraction follows multiplication and is presented as the reverse of addition. Colburn begins by giving five examples which, "though apparently different," all require the same operation-i. e., subtraction. The pupil solves the first examples by using his knowledge of addition.

The operation for the case which requires "borrowing" is pre

sented by writing the numbers thus: 17 is written 10+ 7

Division was considered to be a particularly difficult topic. Colburn starts with some simple problems which he handles in the following fashion:

A boy having 32 apples wished to divide them equally among 8 of his companions. How many must he give them apiece?

probably divide them But to give them one

If the boy were not accustomed to calculating, he would by giving one to each of the boys, and then another, and so on. apiece would take 8 apples, and one apiece would take 8 more, and so on. The question then is, to see how many times 8 may be taken from 32; or, which is the same thing, to see how many times 8 is contained in 32. It is contained four times. Ans.-4 each.

A boy having 32 apples was able to give 8 to each of his companions. How many companions has he?

This question, though different from the other, we perceive is to be performed exactly like it. That is, it is the question to see how many times 8 is contained in 32. We take away 8 for one boy, and then 8 for another, and so on.

A man having 54 cents, laid them all out for oranges at 6 cents apiece. How many did he buy?

It is evident that as many times as 6 cents can be taken from 54 cents, so many oranges he can buy. Ans. 9 oranges.

A man bought 9 oranges for 54 cents; How much did he give apiece?

In this example we wish to divide the number 54 into 9 equal parts, in the same manner as in the first question we wish to divide 32 into 8 equal parts. Let us observe, that if the oranges had been only one cent apiece, nine of them would come to nine cents; if they had been 2 cents apiece, they would come to twice nine cents; if they had been 3 cents apiece, they would come to 3 times 9 cents, and so on. Hence the question is to see how many times 9 is contained in 54. Ans. 6 cents apiece.

In all the above questions the purpose was to see how many times a small number is contained in a larger one, and they may be performed by subtraction. If we examine them again, we shall find also that the question was, in the two first, to see what number 8 must be multiplied by in order to produce 32; and in the third to see what

the number 6 must be multiplied by to produce 54; in the fourth, to see what number 9 must be multiplied by, or rather what number must be multiplied by 9, in order to produce 54.

The operation by which questions of this kind are performed is called division. In the last example, 54, which is the number to be divided, is called the dividend; 9, which is the number divided by, is called the divisor; and 6, which is the number of times 9 is contained in 54, is called the quotient.

Colburn then goes on to tell how to prove division, and following this takes up the case when the combination is not one that has occurred in the multiplication table.

At 3 cents apiece, how many pears may be bought for 57 cents? It is evident that as many pears may be bought as there are 3 cents in 57 cents. But the solution of this question does not appear so easy as the last on account of the greater number of times which the divisor is contained in the dividend. If we separate 57 into two parts it will appear more easy: 57=30+27.

We know by the table of Pythagoras that 3 is contained in 30 ten times, and in 27 nine times. Consequently it is contained in 57 nineteen times, and the answer is 19 pears.

This same method is explained for four more problems, in which he points out how the breaking up of the dividend may be determined. He then continues:

It is not always convenient to resolve the number into parts in this manner at first, but we may do it as we perform the operation.

In 126 days how many weeks?

Operation: 126=70+56. Instead of resolving it in this manner, we will write it down as follows:

I observe that 7 can not be contained 100 times in 126; I therefore call the two first figures on the left 12 tens, or 120, rejecting the 6 for the present. 7 is contained more than once and not so much as twice in 12; consequently in 12 tens it is contained more than 10 times and less than 20 times. I take 10 times 7, or 70, out of 126, and there remains 56. Then 7 is contained 8 times in 56 and 18 times in 126.

The development is continued through four more problems, the last only being abstract and having a divisor of five digits. The rule is then stated, the last thing in the section.

Short division is presented last as a "much abridged" method of performing division when the divisor is a small number.

Fractions arise in examples which require division when there is a remainder. For example, to tell "How many yards of cloth, at 6 dollars a yard, may be bought for 45 dollars" a fraction is necessary. In Sections XII to XXIV, inclusive, except Section XX, common fractions are treated in detail. (See table of contents in Appendix.) A conspicuous feature of this treatment is the departure from the

usual order. It begins with a section in which fractions are manufactured by the pupil in solving such examples as "What part of 7 yards is 4 yards?" "What part of a gallon is a pint?" "What part of 5 dollars is 72 cents?" "What is the ratio of 28 to 9?"

Improper fractions are required to be changed to whole or mixed number in solving such examples as "If a family consume 1/3 of a barrel of flour in a week, how many barrels will last them four weeks? How many will last them 17 weeks?" The reverse operation is required in such as the following: "If 1/15 of a barrel of flour will serve a family one week, how many weeks will 2 4/15 barrels serve them? How many weeks will 18 7/15 serve them?" The multiplication of a fraction by an integer by multiplying the numerator, which comes in the following section, gives exercise upon reducing improper fractions to whole or mixed numbers.

In Section XVI Colburn groups together the division of a number into parts, as, "Bought 43 tons of iron for 4,171 dollars; how much was it a ton?"; and the multiplication of a whole number by a fraction, as, "At $4.20 per box what is the cost of 1/4 of a box of oranges?" These two problems require the same arithmetical operations. In this section are placed such problems as:

If 3 yards 3 qrs. of broadcloth cost $30.00, what will 7 yds. cost?

If the distance from Boston to Providence be 40 miles, how many times will a carriage wheel, the circumference of which is 15 ft. 6 in., turn round in going that distance?

What is 43/53 of a yard?

A merchant bought a quantity of tobacco for $250.00 and sold it so as to gain 3/10 of the first cost; how much did he sell it for?

If 25 men can do a piece of work in 17 days, in how many days will 38 men do it? Three men hired a pasture for $42.00; the first put in four horses; the second, 6; and the third, 8. What ought each to pay?

In these problems are represented the rule of three, descending reduction of compound numbers, profit and loss, discount, and partnership. All of these require nothing more than the division of a whole number into parts or its multiplication by a fraction. The above types of problems are presented with no explanatory notes or headings. That they have to do with a variety of arithmetical topics Colburn is not concerned. But he is anxious that the pupil learn the kinds of situations which call for this operation.

From the standpoint of the mathematician it is interesting to note that Colburn comments upon calling the operation of this section, multiplication, by saying, "Multiplication, strictly speaking, is repeating the number a certain number of times, but by extension it is made to apply to this operation." Division of a fraction by a whole number and multiplication of a fraction by a fraction are presented in Section XVII. In the next section it is pointed out that a fraction may be multiplied by dividing the denominator.

Section XIX has to do with the addition and subtraction of fractions and the necessary reductions to a common denominator and to lower terms. The section contains 32 examples, of which 21 are practical. The drill upon finding the common divisor, least common multiple, and reducing fractions to a common denominator and lowest terms is given in Sections XXI and XXII.

Colburn's approach to reducing fractions to a common denominator is interesting and is eloquent of his general plan to have the pupil see what a process is for before he is asked to perform it.

We observed a remarkable circumstance in the last article, viz, that 1/2=4/8 and 3/7=12/28. This will be found very important in what follows.

A man having a cask of wine, sold 1/2 of it at one time and 1/3 of it at another; how much had he left?

1/2 and 1/3 can not be added together, because the parts are of different values. Their sum must be more than 2/3 and less than 2/2 or 1. If we have dollars and crowns to add together, we reduce them both to pence. Let us see if these fractions can not be reduced both to the same denomination. Now 1/2=2/4=3/6=4/8, etc.

The "remarkable circumstance" had come about from the two ways of multiplying a fraction. Multiplying a fraction by dividing by its denominator gave the result in lower terms than by multiplying the numerator of the fraction.

Sections XXIII and XXIV are devoted to the division of a whole number, or a fraction, by a fraction. After a rather lengthy development this generalization is reached: "Multiply the dividend by the denominator of the divisor, and divide the product by the numerator."

In the next four sections decimal fractions are presented. Their notation is explained as being simply an extension of the decimal system in which a figure has a place value and the topic is treated in Colburn's inductive manner. In general it appears that he believes operations with decimal fractions are similar to operations with whole numbers, and this is the idea he wishes the pupil to get. The only serious difficulty the pupil is going to have, as he sees it, is in division, and he develops this in detail.

The last section is concerning circulating decimals, a topic we did not find in the texts of the previous period. A circulating decimal is one such as arises when one attempts to reduce a common fraction such as 1/7 to an equivalent decimal form. One will get a never ending sequence of figures, but in this sequence certain series of figures will be repeated. After Colburn shows the occasion for circulating decimals, he explains how one may find the equivalent common fraction when they have given a circulating decimal. Except for a list of miscellaneous examples, the text closes with a brief presentation of the proof of multiplication and division by casting

out nines.

Definitions and information given when needed.-We have already pointed out this feature in several instances. It is one of the chief 81758°-17-6