three of these numbers (which occur in the second horizontal row) have a common measure. Again: w' is prime to c, because otherwise w' and c would have a common measure, which-measuring c-would measure x" and y" (multiples of c), as well as w'; so that w', x", and y' would have a common measure, whereas it has been assumed that not more than two of these numbers (which occur in the third horizontal row) have a common measure. And w' is prime to "", both numbers occurring in the last horizontal row. So that w-being prime to b, to c, and to x-is (p. 149, I.) prime to bxcxx". As, therefore, the division of w and x by a gives quotients which are prime to each other, a is (p. 149, II.) the greatest common measure of w and x. Consequently, the least common multiple of w and x-or of a× w' and axbxcxxis (axw')x(axbxcxx"")÷a=axbxc xw' xx". (II.) The greatest common measure of axbx c x w xx" and y-or of axbxcxwxx"" and axbxcxy""is axbx c. For, when we divide the two numbers under consideration by axbxc, we find the resulting quotients, w'xx"" and y"", to be prime to each other. Thus, as w', x'", and y'" all occur in the last horizontal row, y'" is prime to w', and to x"", and therefore to w'xx"". Consequently, the least common multiple of axbxcxw' xx" and y, or of axbxcxwxx"" and axbxcxy"",in other words, the least common multiple of w, x, and y, is (axbx c × w' × x'"') × (axb×cxy"")÷axbxc= axbxcxwxx""' xy"". (III.) The greatest common measure of axbxcx w' x x""xy"" and z-or of a xbx cxw'×x"" × y'" and b × 2'is b. For, dividing the two numbers under consideration by b, we find the resulting quotients, axcxw'xx"" ×y"" and 2', to be prime to each other. Thus, z' is prime to a, because otherwise z' and a would have a common measure, which-measuring a-would measure w, x, and y (multiples of a), as well as a multiple of 2'; so that w, x, y, and z would have a common measure, whereas it has been assumed that not more than three of these numbers (which occur in the first horizontal row) have a common measure. Again: z' is prime to c, because otherwise z F and c would have a common measure, which-measuring c-would measure x" and y" (multiples of c), as well as z'; so that x", y'", and 2' would have a common measure, whereas it has been assumed that not more than two of these numbers (which occur in the third horizontal row) have a common measure. Lastly: z' is prime to w', to x'", and to y"",-the whole four numbers occurring in the last horizontal row. So that z-being prime to a, to c, to w', to x''', and to y""-is prime to axcxwxxxy"". As, therefore, the division of axbxcxwxx""xy"" and z-or of axbxcxwxxxy"" and bx-by b gives quotients which are prime to each other, b is the greatest common measure of axbxcxwxxxy"" and z. Consequently, the least common multiple of axbx cx w' xx" xy" and 2, or of axbxcxwxxxy"" and bx 2′,—in other words, the least common multiple of w, x, y, and 29 is (axbxcxwxx"" xy"") x (bx 2')÷b=axbxcxw' x 2 ax w' axbxcxx"" a xbx cxy""' axbxcxwxx'" bxx FRACTIONAL NUMBERS, OR FRACTIONS. 104. If a unit like one of those which we have hitherto been considering-an INTEGRAL unit, let us now say, for the sake of distinction-were divided into any number of equal parts, the parts, regarded as such, would be FRACTIONAL UNITS. 105. The denomination of a fractional unit is expressed by one of that class of words called "ordinals,"*-the corresponding “cardinal” indicating how many such units there are in an integral unit. Thus, if an integral unit were divided into So that, after eating I fifth of a cake, a child would have 4 fifths remaining-a cake being divisible into 5 fifths; after parting with 3 eighths of his farm, a farmer would have 5 eighths remaining-a farm being divisible into 8 eighths; and, after selling 5 twelfths of a piece of cloth, a draper would have 7 twelfths remaining—the number of twelfths into which a piece of cloth is divisible being 12. 106. Two or more fractional units of the same denomination—that is, two or more "halves," two or thirds," two or more fourths," &c. (as the case may be)-constitute a FRACTIONAL NUMBER, or more " a FRACTION. 66 * There is one exception: instead of "second," we say half. "Three yards," "three shillings," "three fifths," "three eighths:" in each of these instances, the number is "three," whilst the unit is in the first case, a "yard;" in the second, a "shilling;" in the third, a "fifth;" and in the fourth, an "eighth." The first two numbers are "integral;" the last two, "fractional." In practice, we cannot avoid regarding a fractional unit as a fractional "number;" just as we cannot avoid regarding an integral unit as an integral number. For the present, however, it will be well to bear in mind the distinction between a fractional "unit" and a fractional "number." Three fifths (3), four sevenths (4), five ninths (§), &c., are fractional "numbers;" whilst one fifth (3), one seventh (4), one ninth (3), &c., are fractional units." 66 107. In writing a fraction, we employ two numbers, which are placed one below the other, and separated by a horizontal line. Of these two numbers-which are known as the "terms" of the fraction-the upper, called the numerator, indicates how many fractional units the fraction is composed of; whilst the lower number, called the denominator, indicates how many such units there are in an integral unit-in other words, indicates the denomination. Thus 3 fifths; 4 sevenths; 5 eighths; 7 ninths =; &c. A fraction, therefore, which has (say) 10 for numerator is composed of 10 fractional units of the same denomination-10 halves, 10 thirds, or 10 fourths, &c.; but what the denomination is, we are unable to say without knowing the denominator. On the other hand, a fraction which has 10 for denominator is composed of one or more tenths; but the number of tenths we cannot tell without knowing the numerator : 10-10 halves; 10 thirds; 0=10 fourths; =1 tenth; 2=2 tenths; =3_ tenths; 4 tenths; &c. NOTE. If the question were asked, How can there be such a fraction as, for instance, 10 halves of a penny (10d.), the number of halves into which a penny is divisible being only 2? the answer would be this: We can imagine any number of pence divided into halves each, and the division of each of 5 pence into halves would give the fraction referred to; just as we should obtain 11 fourths of a 1-lb. loaf by dividing 3 such loaves into 4 fourths each, and setting aside one of the 12 fourths so obtained. 108. We express a concrete number as a fraction of a larger one of the same kind when-having reduced both numbers to the same denomination— we write the smaller for numerator, and the larger for denominator. Thus, to express 13 grains as a fraction of a pennyweight, we write 13 for numerator, and 24 (the number of grains in a pennyweight) for denominator; to express 2 ft. 5 in. as a fraction of a yard, we write 29 (the number of inches in 2 ft. 5 in.) for numerator, and 36 (the number of inches in a yard) for denominator; to express 3d. as a fraction of 2s. 6d., we write 7 (the number of halfpence in 3d.) for numerator, and 60 (the number of halfpence in 2s. 6d.) for denominator; &c. :13 grs. of a pennyweight; 2 ft. 5 in.=3% of a yard; 3d. of 28. 6d. ; &c. = This requires little explanation. The number of grains in a pennyweight being 24, a grain is the twenty-fourth part of a pennyweight; so that 13 grains must be 13 twenty-fourths (4) of a pennyweight. Again: the number of inches in a yard being 36, an inch is the thirty-sixth part of a yard; so that 29 inches, or 2 ft. 5 in., must be 29 thirty-sixths (%) of a yard. In like manner, the number of halfpence in 2s. 6d. being 60, a halfpenny is the sixtieth part of 2s. 6d. ; so that 7 halfpence, or 3 d., must be 7 sixtieths (7) of 2s. 6d. 109. Of fractional UNITS, the largest is a half (1): of others, a third (3) is larger than a fourth (1); a |