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text. Adams follows the order of Pike, but gives a less elaborate treatment.

Denominate numbers.-Weights and measures were not standardized, and we find a lack of uniformity in the tables of denominate numbers. Dilworth gives the tables of English money, Troy weight, avoirdupois weight, apothecaries' weight, time, and motion (circular measure), in essentially the form we know them to-day. Other systems of measures are given in a form which is only partially like that in our arithmetics to-day, and there are some which have disappeared from our texts. Because of their value in showing a phase of the development of arithmetic, we give the last two classes of tables below:

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A supplementary table to avoirdupois weight is also of interest:

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In addition, in the section on exchange, the tables for the money of a number of foreign countries and even cities are given: Spain, Italy, Venice in Italy, France, Portugal, Florence in Italy, Frankfort in Germany, Antwerp, Brussels, Amsterdam, Rotterdam, Hamburg in Germany, British Dominions in America, the West Indies, Ireland, Denmark, and Stockholm in Sweden.

Pike adds to cloth measure:

6 quarters of a yard make 1 ell-French.

4 quarters, 1 inch and one fifth, make 1 ell-Scotch.

3 quarters and two-thirds make 1 Spanish var.

In long measure, he omits the denomination of hand and adds the surveyors' measure. Square measure is increased by the denominations of square inch, square foot, and square mile, and ale or beer measure by the denominations puncheon and butt. The table of dry measure is as follows:

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Solid (cubic) measure, which Dilworth does not give, is given thus by Pike:

1,728 inches make..

27 feet......

40 feet of round timber, or 50 ft. of hewn timber.....

128 solid feet, i. e., 8 in length, 4 in breadth, 4 in height..

1 foot.
1 yard.

1 ton or load.

1 cord of wood.

The table of Federal money which was established in 1786 is given. by Pike under a section heading, “Decimal tables of coin, weight, and measure." The decimal tables of weight and measure was an attempt to decimalize the tables in common use, though the advantage of the form which he gives is not evident.

In Daboll's text the great majority of the problems are stated in terms of the money of the United States. This is true also in Adams's Scholar's Arithmetic. But Federal money did not become generally used until considerably later than 1800. In 1815 Adams deplores the use of English money and "to shew the great advantage which is gained by reckoning in Federal money" he contrasts "the two modes of account, and in separate columns on the same page," places the same questions "in Old Lawful and in Federal Money."

The simplicity of the decimal system, upon which the Federal money was based, was very soon evident and stood out in contrast to the haphazard basis of the other systems of measure. Chauncey Lee, in 1795, commenting upon "our tables of weight and measure" points out that they "are as illy contrived for ease of calculation as can well be imagined." And later he says:

I am persuaded that experience will soon evince the expediency, if not the absolute necessity of Federalizing all the tables of weights and measures and other mixed quantities, which have an immediate relation to commerce, upon a decimal scale.

After showing the inconvenience of vulgar fractions for the purposes of calculation, he says:

This inconvenience will ever continue to operate in a greater or less degree until this vulgar evil is plucked up by the roots-all these surd, untoward fractional numbers banished from practice, and the several denominations in all commercial tables of mixed quantities conformed to our Federal money and established upon a decimal scale. To accomplish this is a task too great for any individual in a republican government. It requires the arm of Congress to effect it.

He follows this with Federalized tables for avoirdupois weight, troy weight, liquid measure, dry measure, cloth measure, apothecaries' weight, and board measure. His plan involves keeping at least one unit in each table the same except in the case of troy weight. The following table illustrates the plan:

FEDERAL AVOIRDUPOIS.

10 drams make...

10 ounces...

100 pounds.

10 hundreds...

.1 ounce.
1 pound.

1 hundredweight.
1 thousand.

The plan was not adopted, and there is no trace of it in the arithmetics of Daboll and Adams, which appeared a few years later.

The American Accomptant is interesting historically also because it is the first arithmetic in which the dollar mark ($) appears. The mark is in the form. There is also a mark for dimes (), a mark for cents (/), and a mark for mills (/). But these are scarcely used in the text. Daboll gives our present dollar mark, but uses also the abbreviation "dols." He writes both 127 dols., 19 cents, and $381, 72 cents.

Daboll considers "Federal coin" so "nearly allied to whole numbers, and so absolutely necessary to be understood by everyone" that he introduces it immediately following whole numbers. Adams places it after decimal fractions and 45 pages after table of English

money.

The four fundamental operations were usually repeated for denominate numbers under the head of "Compound Addition," "Compound Subtraction," etc. Dilworth divides his treatment of each of the operations into two parts, "simple" and "compound." Pike and Daboll give the operations for "compound" numbers after all operations have been given for "simple" numbers. There are no special rules in Dilworth's text for the operations with "compound" numbers, but other authors usually give specific rules. Pike recognizes as many as eight cases of "Compound Multiplication." Reduction, ascending and descending, was an important topic in the texts.

1 For a discussion of the origin of the dollar mark, see F. Cajori: The Evolution of the Dollar Mark, Popular Science Monthly, vol. 81, p. 521.

It occupies 10 pages in Dilworth's text, which marks it as one of the most important topics-addition, practice, and exchange being the only ones which are given more space.

Rule of three.-There are three cases of the rule of three which Pike defines as follows:

The Single 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.

The Single Rule of Three Inverse teacheth, by having three numbers given, to find a fourth, which shall have the same proportion to the second, as the first has to the third.

The Double Rule of Three teacheth to resolve such questions as require two, or more, statings by simple proportion; and that, whether direct or inverse. It is composed (commonly) of 5 numbers to find a sixth, which if the proportion be direct, must bear such proportion to the fourth and fifth as the third bears to the first and second: but if inverse, the sixth number must bear such proportion to the fourth and fifth as the first bears to the second and third.

For centuries this rule was the basis of the rules for solving most of the problems arising in business. Its application was made so universal that it was often spoken of as "The Golden Rule" of arithmetic. We shall describe the three forms of the rule and then illustrate the variety of practical situations to which it was applied in the arithmetics of this period.

The rule given for the case of direct proportion was to pick out "the number that asks the question" for the third term, take the one of the "same name or quality" for the first term, and the remaining one which has the same name or quality as the required answer is the second term. The solution is then accomplished by multiplying the second and third terms together and dividing by the first, the quotient being the answer.

The problems under this rule were of the type: "If 6 lbs. of sugar cost 9s., what will 30 lbs. cost at the same rate?" This type of problem was often complicated, as when the first and third terms were not of the same denomination, or when a term was expressed in more than one denomination. Pike recognizes seven cases of these complications for which he gives special directions.

Problems requiring the rule of three inverse are to be distinguished from those belonging to the direct case—

by an attentive consideration of the sense and tenor of the question proposed: for if thereby it appears that when the third term of the stating is less than the first, the answer must be less than the second, or when the third is greater than the first, the answer must be greater than the second, then the proportion is direct; but, if the third be less than the first, and yet the sense of the question requires the fourth to be greater than the second, or if the third being greater than the first, the answer must be less than the second, the proportion is inverse.

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