The entire matter of long multiplication with integers and United States money should be treated as a teaching unit and the bonds formed in close organization, even though numbers as large as 900,000 are occasionally involved. The reason is not that it is more logical, or less scrappy, but that each of the bonds in question thus gets much help from, and gives much help to, the others. In sharp contrast to a topic like 'long multiplication' stands a topic like denominate numbers. It most certainly should not be treated as a large teaching unit, and all the bonds involved in adding, subtracting, multiplying, and dividing with all the ordinary sorts of measures should certainly not be formed in close sequence. The reductions ascending and descending for many of the measures should be taught as drills on the appropriate multiplication and division tables. The reduction of feet and inches to inches, yards and feet to yards, gallons and quarts to quarts, and the like are admirable exercises in connection with the (_a_ × _b_) + _c_ = .... problems,--the 'Bought 3 lbs. of sugar at 7 cents and 5 cents worth of matches' problems. The reductions of inches to feet and inches and the like are admirable exercises in the _d_ = (.... × _b_) + _c_ or 'making change' problem, which in its small-number forms is an excellent preparatory step for short division. They are also of great service in early work with fractions. The feet-mile, square-foot-square-inch, and other simple relations give a genuine and intelligible demand for multiplication with large numbers. Knowledge of the metric system for linear and square measure would perhaps, as an introduction to decimal fractions, more than save the time spent to learn it. It would even perhaps be worth while to invent a measure (call it the _twoqua_) midway between the quart and gallon and teach carrying in addition and borrowing in subtraction by teaching first the addition and subtraction of 'gallon, twoqua, quart, and pint' series! Many of the bonds which a system-made tradition huddled together uselessly in a chapter on denominate numbers should thus be formed as helpful preparations for and applications of other bonds all the way from the first to the eighth half-year of instruction in arithmetic. The bonds involved in the ability to respond correctly to the series:-- 5 = .... 2s and .... remainder 5 = .... 3s and .... remainder 88 = .... 9s and .... remainder