Pike's arithmetic is an elaborate treatise and not a text for the use of young pupils. It represents the maximal content of arithmetic in this period. The book sold for $2.50, which placed it out of the reach of many pupils. It was used primarily in academies and colleges and yet it had a considerable circulation. A second edition was printed in 1797, a third in 1808, a fourth in 1822, and a fifth in 1832. An abridged edition was published in 1793, and a second one, prepared by Dudley Leavitt, appeared in 1826.

Following 1788, texts by American authors appeared with increasing frequency. The American Tutor's Assistant, by Zachariah Jess, 1798; The Schoolmaster's Assistant, by Nathan Daboll, 1799; A New System of Mercantile Arithmetic, by Michael Walsh, 1800; Scholar's Arithmetic, by Daniel Adams, 1801; and Scholar's Arithmetic, by Jacob Willetts, 1817, were widely used.

Of these texts, Daboll's Schoolmaster's Assistant seems to have been most popular. An edition "improved and enlarged,' was published as late as 1839. Adams's Scholar's Arithmetic had passed through 9 editions, and 40,000 copies had been sold when it was revised in 1815. An edition was published in 1822. Jacob Willetts's Scholar's Arithmetic passed "through more than 50 editions in a few years." A revised edition was published in 1849. A third revised edition of 20,000 copies of Walsh's Mercantile Arithmetic was printed in 1807. An edition was published as late as 1826.

The content of the texts. Since Dilworth's Schoolmaster's Assistant was the first text in arithmetic to attain an extended circulation in this country, it will be used as a basis for an exposition of the content of the texts of this period. Reference will be made to features of other popular texts which were significant.

The theory of arithmetic.-Theoretical arithmetic was recognized in the definitions of arithmetic which were given in these early texts. The space given to arithmetical theory varied. Dilworth's text is primarily a practical arithmetic and he gives very little in the way of demonstrating "the reason of practical operations," and he has nothing to say about "the nature and quality of numbers." Pike attempts to treat comprehensively both theoretical and practical arithmetic. The spirit of the mathematician who is interested in the theory of numbers and operations pervades the whole book. In footnotes he demonstrates the operations. Under the head of "Vulgar Fractions" he defines prime number, composite number, and perfect number, and gives 10 perfect numbers which he states are "all which are, at present known." The other texts of the period show much less emphasis upon arithmetical theory. Often considerable space was given to a "demonstration" of the rules, but these demonstrations were usually explanations of the application of a rule to a particular problem or example.

Definitions. The definitions of number, fraction, addition, etc., were usually given in an abstract form, with no reference to the concrete situations which required the arithmetical concept or operation. For example, addition was defined as "putting together two or more numbers or sums, to make them one total, or whole sum." In the case of business rules, an attempt was made to indicate the sort of situation which called for the particular rule. But the practical situation itself was not described except in the problems. There was usually no attempt to build up a logical system of definitions. Notation and numeration.-Dilworth made this topic, which he styles, "Notation," the first in the text after some preliminary definitions. Numeration consisted of rules for reading numbers, and they are given for reading numbers up to 9 digits. Pike's rule extends to sextillion, 42 digits, and in a note to duodecillion, 78 digits. The periods are of 6 digits each. Daboll also used 6 digits to a period, and he gives four such periods. "Notation of numbers by Latin letters" is mentioned, but not given by Dilworth. Wingate gives Roman numerals and prefers IIII to IV, VIIII to IX, etc., and IIX is given with VIII for eight. Pike gives Roman notation, but Daboll and many other authors omit it.

The fundamental operation for integers.-These operations were given in the serial order, addition, subtraction, multiplication, and division. Sometimes this order was interrupted to give the tables of denominate numbers after addition. This is the case in Dilworth's text. In addition he gives the rule for placing the numbers to be added and recommends proving by adding in reverse order. He does not mention "carrying" and solves out no examples. Nine abstract examples are followed by 15 pages of "compound" addition. The rule for subtraction is given, but otherwise the presentation is similar to that of addition. In multiplication, the tables are given from 3 to 12 inclusive, except the tens. The process of multiplication is given in five cases: First, when the multiplier is 12 or less; second, when the multiplier "consists of more figures than one"; third, when the factors "have cyphers at the ends"; fourth, when the multiplier has cyphers "between the significant figures"; fifth, when the multiplier may be resolved into two factors, each being less than 10. Short division is disposed of with no rule and only 12 examples. Long division is taken up in three cases, with a rule for each: First, any divisor; second, when there are cyphers at the end of the divisor; third, when the divisor "is such a number" that it is the product of "any two figures.' In no case is an example worked out as a model or the rule explained. Besides each operation being applied to "compound numbers," there is also a list of practical problems for each rule.

In other texts the fundamental operations are presented in a more simplified form. Cocker, in general, explains a process before he applies it to a particular example. Hodder carefully explains an example, even in addition, before he states the rule. Pike and Daboll give addition and subtraction tables. Most authors give the table of Pythagoras. Pike "demonstrates" the rule for multiplication and division. Cocker and Hodder attempt to add to the understanding of multiplication and division by telling of the situations which require the operations. Hodder speaks of multiplication as being equal to many additions. Daboll says "division is a concise way of performing several subtractions," The forms of the operations are essentially the same as our present forms with one or two exceptions in the older English texts.

In addition to the five cases of multiplication given by Dilworth, Pike recognizes the seven following cases: First, to multiply by 10, 100, 1,000, etc.; second, "to multiply by 99, 999, etc., in one line"; third, "to multiply by 13, 14, 15, etc., to 19, inclusively, at one multiplication"; fourth, "to multiply by 111, 112, 113, to 119, so as to have the product in one line;" fifth, "to multiply by 101, 102, 103, etc., to 109, so as to have the product in one line;" sixth, "to multiply by 21, 31, 41, etc., to 91, in one line;" seventh, "to multiply by 22, 23, 24, etc., to 29, so as to have the product in one line." In addition to these 12 cases a general rule is given for multiplying "any number, viz, whole or decimal, by any number, giving only the product." Detailed specific rules are given for each case; for some cases two such rules are given. But there is a marked tendency in the texts after Pike's in the direction of fewer cases. Daboll recognizes only five cases and Adams gives besides the general rule only a section to "contractions and varieties in multiplication.'

A knowledge of the addition and subtraction facts seems to have been taken for granted. Some of the texts do not give an addition or subtraction table. The multiplication and division tables are usually given and were to be memorized. Adams says under multiplication, "Before any progress can be made in this rule, the following table must be committed perfectly to memory." There are no exercises to be solved orally, and there is no provision for drill upon the number facts contained in the tables.

Common, or vulgar fractions.-Dilworth devotes Part II of his text to vulgar fractions (see Appendix). Following the definition of a fraction as "any two numbers placed thus, ," and the definition of terms and the "sorts of vulgar fractions," reduction of fractions is given in 12 cases. They are: (1) Reduction to common denominator; (2) reduction to lowest terms; (3) and (4) reduction of “mixt" number, to improper fraction and reverse; (5) reduction of compound fraction to a single fraction; (6) to reduce a fraction of one

A form of the multiplication table.

denomination to, a fraction of another, but greater; (7) to reduce a fraction of one denomination to a fraction of another but less; (8) to "reduce vulgar fractions from one denomination to another of the same value, having the numerator of the required fraction given"; (9) the same except the denominator of the required fraction is given; (10) to reduce "a mixed fraction to a single one"; (11) to "find the proper quantity of a fraction in the known parts of an integer"; (12) "to reduce any given quantity to the fraction of any greater denomination of the same kind." The operations of addition, subtraction, multiplication, and division for fractions are then disposed of within three scant pages. Two pages devoted to the single rule of three direct, single rule of three inverse, and double rule of three for vulgar fractions complete Part II. For each of the four operations a specific rule is given, e. g., the rule for multiplication is, "Multiply all the given numerators for a new numerator, and all the denominators for a new denominator."

For reducing a fraction to its lowest terms, Dilworth gives only the Euclidean process. In general the other authors give the rule, "Divide the terms of the given fraction by any number which will divide them without remainder, and the quotients, again, in the same manner; and so on till it appears that there is no number greater than 1 which will divide them." Pike and Daboll give both methods. Dilworth's rule for reducing fractions to a common denominator is: "1. Multiply each numerator into all the denominators but its own for a new numerator. 2. Multiply all the denominators for a new denominator." The least common denominator is not mentioned, although it would be very useful in the examples he gives. Pike and Daboll give in addition the method for reducing to a least common denominator.

Dilworth does not solve an example or illustrate a rule. Cocker and Hodder and the later authors, in general, solve out one example under a rule and usually carefully explain the operation. Wingate suggests cancellation as a short method in multiplication of fractions. Daboll also does this. Pike gives three cases under multiplication.

The contrast in the position and space given to common fractions is interesting. Hodder and Pike place them immediately following denominate numbers and reduction. Daboll gives three cases of reduction of fractions immediately following denominate numbers, but the real treatment of the topic comes nearly 100 pages later in the text. Adams finishes with fractions with a scant page devoted to explaining the meaning of a vulgar fraction and closes by saying: "The arithmetic of vulgar fractions is tedious and even intricate to beginners. We shall not therefore enter into any further consideration of them here."

Dilworth and Daboll make no attempt to explain the meaning of a fraction. They just tell what the symbol is and how it is to be

operated upon. Adams gives two illustrations to explain the meaning of a fraction. The examples are abstract, the nearest approach to a practical problem being in such as: "Add 4 of a yard, of a foot, and of a mile together." Factoring, highest common divisor, and least common multiple are not mentioned by Dilworth. Pike gives them as the first topics under the head of fractions.

Vulgar fractions were even omitted in a few texts. Chauncey Lee in The American Accountant, 1797, explains his reason for omitting them as follows:

As the use of vulgar fractions may be advantageously superseded by that of decimals, they are viewed as an unnecessary branch of common school education and therefore omitted in this compendium.

Decimal fractions.-Part III of Dilworth's Schoolmaster's Assistant, which bears the title, "Of Decimal Fractions," includes much subject matter which is not commonly included under this head. Besides notation, reduction, addition, subtraction, multiplication, and division for decimals, the section contains evolution, the rule of three, interest, discount, equation of payments, and a number of other applications of percentage. (See Appendix.) The four operations are presented very briefly and entirely abstractly. Reduction includes such examples as, "Reduce 76 yards to a decimal of a mile,” and the reverse exercise.

The place occupied by decimal fractions in this text is significant of the esteem in which they were held. As compared with common fractions, the rule of three, interest, partnership, and other topics, decimal fractions were new. The elementary arithmetical processes, with the exception of decimal fractions and logarithms, were matured by the close of the sixteenth century. Simon Stevin gave the first systematic treatment of decimal fractions in 1585, and their application to practical arithmetic was a contribution of the seventeenth century. Coming thus after methods for the calculations of business had been worked out, which were moderately satisfactory, decimal fractions and the methods of calculation which they make possible were incorporated in the texts only very slowly. Hodder, 1661, does not mention them in his table of contents, but approaches them in a chapter on profit and loss. Dilworth, as we have seen, treats all of the more common problems of business before he mentions decimal fractions. This shows that a need for them was not keenly felt.

The establishment of a Federal money, 1786, increased the usefulness of decimal fractions and marked the beginning of their increased importance as a topic of arithmetic in the United States. Pike, who gives a brief account of Federal money immediately after decimal fractions, places them early in his text. Daboll places Federal money immediately after addition of integers, but the position and treatment of decimal fractions is essentially the same as in Pike's