same rule. For days and months aliquot parts of a year were to be taken. For 6 per cent a special rule was given.

Dilworth treats briefly compound interest and rebate or discount (true discount) in Part I. Later in Part III these topics, together with other applications of percentage, are taken up with decimals. Under the head of "Simple Interest" "the ratio of the rate per cent" is defined as "only the simple interest of 1 L. for one year at any proposed rate of interest per cent." It is to be found by the application of the rule of three thus:

£ £ 1

100: 6: 1: 0.06

A table of ratios and the four cases of interest are given, the rule being stated only in term of a formula. No problem is solved out, but presumably decimal fractions are to be employed in solving the problems. Compound interest is presented in the same manner. Annuities and pensions in arrears, present worth of annuities, annuities and leases, and rebate or discount are considered for both simple and compound interest. With only a very few exceptions all possible cases are given. Besides these the topics of purchasing freehold or real estates and purchasing freehold estates in reversion are treated in their several cases. In all cases the rule is stated in terms of a

formula.

Pike adds commission, brokerage, partial payments, buying and selling stocks, and policies of insurance as applications or phases of interest. These topics are treated very simply with the exception of policies of insurance, which is given in eight cases. Four of these cases have to do with problems arising in marine insurance.

Both Dilworth and Pike give equation of payments by the common way and by the true way. By the common way the equated time of payment was found by multiplying "each payment by the time at which it" was due and then dividing "the sum of the products by the sum of the payments." The rule for the true way is complicated, but it is based on the recognition that to be absolutely fair interest upon the amounts whose payment is delayed should be equal to the (true) discount upon the amounts which are paid before they are due. Daboll and Adams do not mention equation of payments by the true way. Adams gives a Massachusetts rule for partial payments, and Daboll adds the Connecticut rule.

In the treatment of these several topics which we now associate under the head of the applications of percentage, decimal fractions are used only as a second method. Six per cent always stood for at the rate of 6 L. on 100 L., $6 on $100, 6 cents on 100 cents, etc. To get from 6 per cent to .06 the rule of three was required, and then .06 was called the ratio. In the general organization of the texts after

Dilworth, decimal fractions were placed early enough so that they might have been used directly in the solution of problems which involved "per cent," but in general they were not. The method of solution was accomplished by an application of the rule of three, or directions were given to divide the product by 100 or to cut off two places.

Percentage with its several cases is not contained in the texts. The range of application is as great as we have to-day, but they were handled without the technique of percentage.

Tare and trett had to do with rules for making allowances in the weight of merchandise. Tare was an allowance for the container (box, barrel, bag, etc.). Trett was an allowance of 4 pounds out of each 104 pounds for "waste and dust in some sort of goods." Cloff was an additional allowance of 2 pounds on every 3 hundredweight. Progressions, arithmetical and geometrical (sometimes called proportion) are treated exhaustively by Pike and partially by Dilworth and Daboll and not at all by Adams. In geometrical progressions the problems are mostly concerning a crafty person who makes an apparently foolish bargain. It involves a geometrical progression, however, and turns out to be most profitable. The merchant who sold 39 yards of fine velvet trimmed with gold at 2 pins for the first yard, 6 pins for the second, 18 pins for the third, etc., is typical.

Permutations was frequently given as a topic. The problems are about such questions as the number of changes which can be rung on a chime of bells, or how many different positions a party can. assume at a dinner table. Pike asks how many variations can be made of the alphabet. In Pike's text the topic of combinations is added, and the whole topic elaborated into seven cases. However, the topic is usually very briefly treated.

Evolution.-Dilworth disposes of square root by saying to prepare the given square for extraction "by pointing off every two figures." He gives no further instructions. Cube root he explains in some detail, employing the relation (a+e)3 = a3 +3a2e + Sae2 + e3, and rules are given for finding all roots up to and including the twelfth root. Pike gives an additional method for cube root and a general rule for "extracting the roots of all powers," but does not explicity go beyond the fifth root. He gives also a general method by approximation. Adams considers only square and cube roots. For these he gives elaborate demonstrations which are illustrated by cuts.

Longitude and time. This had not yet become a topic in the texts of this period. Pike approaches the topic in four problems under the rule of three and in two problems under duodecimals.

Mensuration. The space given to mensuration varied. Dilworth does not mention it. Pike makes it quite a feature. He introduces

practically all of the rules of geometry. (See table of contents in Appendix.) In the other texts it is usually mentioned. The mensurational problems are from commerce rather than from the trades. Duodecimals.-Part V of Dilworth's text is devoted to duodecimals or cross multiplication. In the preface he states that the topic was not contained in the original text, but added in a revision. Duodecimals are defined as "fractions of a foot, or of an inch, or any part of an inch having 12 for their denominations." Feet, inches, seconds, thirds, and fourths are used. It was the purpose to use a scale of 12 in calculations rather than the decimal scale upon which our number system is based. Other texts of the period give the topic; and the system seems to have been used in practical calculations. Adams speaks of it as a rule which is "particularly useful to workmen and artificers in casting up the contents of their work."

Pleasant and diverting questions.-Arithmetical puzzles are occasionally found mixed in with practical problems. In addition, some authors give a list of puzzles under the above or a similar title. This is the case in the texts by Dilworth and Adams and in their lists we recognize some familiar friends from which we select the following: Says Jack to his Brother Harry, I can place four threes in such a manner that they shall make just 34; can you do so too?

As I was going to St. Ives,

I met seven wives,

Every wife had seven sacks,
Every sack had seven cats,
Every cat had seven kits,
Kits, cats, sacks and wives,

How many were going to St. Ives?

Three jealous husbands with their wives, being ready to pass by night over a river, do find at the waterside a boat which can carry but two persons at once, and for want of a waterman they are necessitated to row themselves over the river at several times: The question is, how shall these persons pass 2 by 2, so that none of the three wives may be found in the company of one or two men, unless her husband be present?

Proofs.-Dilworth gives what he calls a "proof" for many of the rules of his text. But these "proofs" are rather checks upon the operations than a proof of the rule. A very common form of proof was to reverse the order, i. e., take the answer obtained and work back to the conditions of the problems. In the case of addition and multiplication, it meant to change the order of performing the operation. "Casting out the nines" was used as a method of proving multiplication and addition, but Pike says, following his exposition of this method of proof:

However, the inconveniency attends this method, that, although the work will always prove right, when it is so; it will not always be right when it proves so; I have therefore given this demonstration more for the sake of the curious, than for any real advantage.

Types of problems.-A very small per cent of the total number of problems call for the arbitrary manipulation of abstract numbers, such as, "Multiply 4786 by 753," or, "What is 14 per cent of 8392?” This is due in part to the relatively small space given to the fundamental operations and to the absence of provision for drill upon special rules, such as percentage. There are a number of problems of this type: "Reduce 16 miles to barleycorns." Such problems are essentially abstract, even though they have to do with concrete quantities. They lack the setting of a practical situation which calls for the operation. Instead, the operation is dogmatically demanded. Furthermore, the problems of this type were not always constructed so as to conform to the demands which are made by actual practical situation. For example, it is difficult to imagine the practical situation which would demand the above process.

A somewhat different type of problem, but being practical only in a slightly greater degree, is the following:

From the Creation to the departure of the Israelites from Egypt was 2,513 years; to the siege of Troy, 307 years more; to the building of Solomon's temple, 180 years; to the building of Rome, 251 years; to the expulsion of the kings from Rome, 244 years; to the destruction of Carthage, 363 years; to the death of Julius Cæsar, 102 years; to the Christian era, 44 years; required the time from the Creation to the Christian era?

While such a problem might arise, it is unusual, and the way in which it is stated leaves the pupil to construct the situation which would give rise to such a problem.

A large number of problems, in many texts a majority, were practical in the sense that the statement of the problem included a description of the practical situation which demanded the calculation. But because of the excessive classification of problems under particular rules, the pupil did not need to use his understanding of the practical situation to determine the operations which were required.

The method of presenting a topic. The manner in which the authors introduced the pupil to arithmetic is typical of the spirit of the texts. In Daboll's Schoolmaster's Assistant, which was probably more extensively used in the United States after 1800 than any other arithmetic before Colburn's, the pupil was introduced to the subject as follows:

Arithmetic is the art of computing by numbers, and has five principal rules for its operation, viz, numeration, addition, subtraction, multiplication, and division. Numeration is the art of numbering. It teaches to express the value of any proposed number by the following characters or figures:

1, 2, 3, 4, 5, 6, 7, 8, 9, 0 or cypher.

Besides the simple value of figures, each has a local value, which depends upon the place it stands in, viz, any figure in the place of units represents only its simplest value, or so many ones, but in the second place, or place of tens, it becomes so many tens, or ten times its simple value.

In the Scholar's Arithmetic, by Daniel Adams, the subject was begun as follows:

Arithmetic is the art or science which treats of numbers.

It is of two kinds, theoretical and practical.

The theory of arithmetic explains the nature and quality of numbers, and demonstrates the reason of practical operations. Considered in this sense, arithmetic is a science.

Practical arithmetic shows the method of working by numbers so as to be most useful and expeditious for business. In this sense arithmetic is an art.

There are six pages of definitions of this sort, and an explanation of the system of notation, before any problems are given. Addition is begun with the definition, followed by the rule for addition and for proving the work. The first example is: "What will be the amount of 3612 dolls. 3043 dolls. 651 dolls. and of 3 dollars when added together?" There is nearly a page of explanation. This is followed by nine abstract examples which complete the topic of addition except for a "Supplement to Addition," which was added in the revised edition.

In treating a topic four elements were recognized-definitions, rule, explanation, and problems. If the topic permitted being subdivided into cases, this was done. The presentation of the single rule of three direct in Pike's text is perhaps typical.

The Rule of Three Direct teacheth, by having three numbers given, to find a fourth that shall have the same proportion to the third as the second hath to the first. If more require more, or less require less, the question belongs to the Rule of Three Direct.

But if more require less, or less require more, it belongs to the Rule of Three Inverse. Rule. 1. State the question by making that number which asks the question the third term, or putting it in the third place; that which is of the same name or quality as the demand, the first term; and that which is of the same name or quality with the answer required, the second term.

2. Multiply the second and third numbers together, divide the product by the first, and the quotient will be the answer to the question, which (as also the remainder) will be in the same denomination you left the second term in, and which may be brought into any other denomination required.

Two or more statings are sometimes necessary, which may always be known from the nature of the question.

The method of proof is by inverting the question.

But, that I may make the method of working this excellent rule as intelligible as possible to the learner, I shall divide it into the several cases following:

1. The fourth number is always found in the same name in which the second is given, or reduced to; which, if it be not the highest denomination of its kind, reduce to the highest, when it can be done.

2. When the second number is of divers denominations, bring it to the lowest mentioned, and the fourth will be found in the same name to which the second is reduced, which reduce back to the highest possible.

3. If the first and third be of different names, or one or both of divers denominations, reduce them both to the lowest denomination mentioned in either.

4. When the product of the second and third is divided by the first; if there be a remainder after the division, and the quotient be not the least denomination of its