was $42-07; against the shoemaker, $39.39. Who received balances in cash? Ans. Butcher received from tailor, 66 Tailor $19.77. NOTE.-By an Act of Congress passed Feb. 20, 1849, the Double Eagle and the Gold Dollar were added to the Gold Coins of the United States. The act directing the coinage of these pieces is to be in force until March 4, 1851. See Art. 58. DENOMINATE NUMBERS. 64. A SIMPLE NUMBER is an expression for a certain number of units having no reference to particular things. Thus 37 is the same as 37 times one, abstractly considered; that is, considered apart from anything that units might represent. It does not mean 37 times a pound, foot, dollar, or anything else. A DENOMINATE NUMBER is an expression for a certain number of units having reference to particular things. It denominates what things are meant. Thus 8 yards is a denominate number whose unit is 1 yard; 8 dollars is a denominate number whose unit is one dollar. Several numbers of different denominations are frequently grouped together, as 6 feet 3 inches. All our different kinds of weights and measures are denominate numbers. It is much to be regretted that we are obliged to employ such a variety of different measures, when the same end would be accomplished by one measure for weight, and one for each of the three geometrical magnitudes, lengths, surfaces and solids,and one for time The French government have adopted such a system of weights and measures, graduated on the decimal scale of notation. In multiplication, the multiplicand being repeated a certain number of times, or a certain fraction of a time, when the multiplier is a fraction, it follows that the multiplier, considered as a multiplier, must always be regarded as a simple or abstract number. And since the product is a repetition of the multiplicand, it must be like the multiplicand; that is, if the multiplicand is an abstract number, the product will be an abstract number; if the multiplicand is a denominate number, the product will be a denominate number of the same kind. In division, the quotient showing how many times the divisor is contained in the dividend, or what fraction of a time when the divisor is greater than the dividend, it follows that the quotient must be regarded as an abstract number; and that the divisor and dividend must be alike. NOTE. In many cases, however, the process of division is rather the dividing of a dividend into as many equal parts as are indicated by the divisor; in which case, the quotient, expressing the units in one of those parts, is of the same kind as the dividend, while the divisor is to be regarded as an abstract number. See Example, Art. 86. What is a simple number? What is a denominate number? What kind of numbers are all our different weights and measures? What is said of the French measures? In multiplication, can the multiplier ever be a denominate number? Are the product and multiplicand always alike? What is said of the quotient? What is said in the note ? The following are some of the most important tables of weights and measures at present employed in this country. ENGLISH MONEY. 65. The denomination of English money are Farthings, Pence, Shillings, and Pounds. The pound sterling, which was not a coin, but a bank note of 20 shillings, has now gone into disuse, and a gold coin, called a Sovereign, supplies its place; but the name pound is still given to 20 shillings. NOTE.-Farthings are sometimes expressed in fractions of a penny, as follows: 1 farthing=d., 2 farthings=d., 3 farthings = d. What are the denominations of English money? Which denomination is never coined? What gold coin is equivalent in value to one pound? Repeat the Table. How are farthings sometimes expressed? TROY WEIGHT. 66. The original of all weights used in England was a grain or corn of wheat, gathered out of the middle of the ear; 32 of these, well dried, were to make one pennyweight, 20 pennyweights one ounce, and 12 ounces one pound. But in latter times, it was thought sufficient to divide the same pennyweight into 24 equal parts, still called grains, being the least weight now in common use. Coins, precious metals, jewels, and liquors, are weighed by Troy weight. * The full weight and value of English gold and silver coin are as in the succeeding table, note. TABLE.* 24 grains gr. make 1 pennyweight, pwt. 12 ounces 66 1 pound, 02. lb. What was the original of all weights used in England? How was the grain ob tained? Is it still used as a weight? What substances are weighed by Troy weight? Repeat the Table, APOTHECARIES' WEIGHT. 67. This weight, as its name would imply, is used in weighing medicines in small quantities, as for prescriptions. But drugs and medicines in gross are bought and sold by Avoirdupois Weight. The pound and ounce Apothecaries' Weight are the same as in Troy Weight. *This scale of weights is said to have been borrowed originally from Troyes in France-hence its name. Some, however, contend that the name has reference to the monkish title given to London, of Troy Novant, |