An Inductive Arithmetic. Notation and Numeration. A certain table is 12 feet long. A second table is 4 feet long. The length of the first table is how many times that of the second? The surface of the first table is 60 square feet, and of the second table 12 square feet. The surface of the first table is how many times that of the second? The volume of a certain block of wood is 8 cubic feet. The volume of a second block is 2 cubic feet. The volume of the first block is how many times that of the second? The weight of the first block is 500 pounds, and of the second block 100 pounds. The weight of the first block is how many times that of the second? A boy can walk a certain distance in 25 minutes. He can ride the same distance on his bicycle in 5 minutes. His speed on the bicycle is how many times his speed on foot? A man pays for a carriage $90 in ten-dollar bills. How does the value of the carriage compare with the value of each of the bills? A certain bridge will sustain a weight of 20 tons. A second bridge will sustain a weight of only 5 tons. The strength of the first bridge is how many times that of the second? Anything that can be measured, or definitely compared, is called Quantity. Observe that the quantities we have compared are, In the first problem, extensions in one direction, or Length. In the second problem, extensions in two directions, or Surfaces. In the third problem, extensions in three directions, or Volumes. In the fourth problem, the comparative effects of the attraction of the Earth on two bodies, or Weights. In the fifth problem, the durations of Time required by the two methods of locomotion. In the sixth problem, the Values of the two articles, as determined by the relative desires for each of them by the buyer and seller. In the seventh problem, the Capacities of the two bridges to resist the effect of gravity on bodies supported by them. NOTE 1. The first six quantities, Length, Surface, Volume Weight, Time and Value, are the ones most commonly met with in arithmetical computations. Among the other capacities which can be measured by their effects, are the strength of an electric current, which is measured in volts and amperes; the power of a steamengine, which is measured by horse powers, and the intensity of lights, which is measured by candle-powers. NOTE 2. It might seem necessary to include gallons, bushels, etc., in a separate class. A little reflection, however, will show that these quantities fall under one of the first three classes. Under which class do they fall and why? About how long is your school-room? About what quantity of boards is there in its floor? (To measure a surface multiply its length by its breadth.) About what quantity of air does it contain? (To measure a volume multiply together length, breadth, and thickness.) About what is the weight of this air? (It takes about 12 cubic feet of air to weigh a pound.) About how long does it take you to walk from your home to your school-room? About what is the value of the taxable property of your town? What standard quantity, in common use and established by law, do we employ in the first of the preceding problems? in the second? in the third? in the fourth? in the fifth? in the sixth? A quantity employed to measure another is called the Unit of the measured quantity, and the process of measuring consists in finding, directly or indirectly, how many, or the Number of, times the unit can be applied. Hence the following definitions: Quantity is anything that can be definitely measured. (Name four quantities.) A Unit of Quantity is a quantity employed as a standard of measurement. (Name the unit that would be employed in measuring each of the four quantities you have selected.) Number is the answer to the question, How many? (Name the number you would obtain in measuring each of the four quantities you have selected.) Arithmetic is the science of numbers. Numbers of One Figure. 1. Numbers to ten are expressed in Arithmetic as follows: four five six seven eight nine one two three 0 1 2 3 4 5 6 7 8 9 The characters 0, 1, 2, 3, 4, etc. are called Figures; the numbers they represent when standing alone are called Units. The character 0, which represents nothing in itself, is called Zero, and, in distinction, 1, 2, 3, etc., which signify value though standing alone, are called Significant Figures. The process of expressing numbers in figures is called Notation. When numbers are expressed in figures, the process of reading them or writing them in words is called Numeration. Numbers of One Period. 2. Groups of units from nine through nine hundred ninetynine, and parts of units from one thousandth through nine hundred ninety-nine thousandths, are expressed in accordance with the following |