case, if the pupil does not understand the principle after rationally using it, any formal "demonstration " is a mere delusion; for any so-called demonstration is grounded on the principle-in general is the principle -merely illustrated or used in a disguised form. For example: Prove $= $15. Since $3 = $1 × 3: multiply by 4, and we have $X 4 = $1 × 4 × 3 = $3; multiply these equals by 5: ... $$ × 4 × 5 = $3 × 5 = $15; but also $15 × 20 = $15; ... $$ × 20 = $1 × 20; dividing equals by equals; na × nb = na; but ne nb = na; nb b = a; multi &nb = × nb; dividing both sides by nb; na These and similar proofs are in essence the idea already considered that if a quantity is divided into a certain number of equal parts, each part has a certain value; if into twice the number of parts, each part has half the value of the former part; if into three times as many parts, each has a third of the value; if into n times as many parts, each has 1 nth of the value. If formal proof is wanted of this important principle (which is, once more, the principle of number), the following is perhaps as intelligible as any other. To prove, for example, that $4= $37 = $15% =, etc., we have $4 = $38 = $188, =, etc.; 75 100 4. The Fraction as Division.-While in its primary conception the fraction is not simply a formal division, it nevertheless involves the idea of division, and can not be fully treated without identifying it with the formal process. The quantity foot, first regarded as foot X 7, must be recognized in its psychological correlate, 7 feet X-i. e., 7 feet 12. As has been shown more than once, these measurements can not but be recognised as two phases of the same measurement, whenever the process becomes the object of conscious attention. It is the law of commutation, the connection between the number and the magnitude of the units in a measured quantity. If we do not know that of 3 times a quantity is 3 times of the quantity— or, generally, that of n times q = n times of q-we have no clear conception of number. If a quantity is measured by of a certain unity of reference taken 7 times, this is seen to be identical with of one of these unities of a second + of a third .; that Numerous illustrations and so-called proofs may be given. Examples: is, in all, of seven of them. (1.) Show that of any quantity is equal to of 3 each measured quantities equal to A B. It is obvious that A K, which is of A B, is equal to A G+CH+ EL; that is, equal to of 3 times A B. (2) To show that foot X 77 feet X-i. e., 7 feet 12. The unit of reference, 1 foot, may be thought of and expressed as 12 twelfths foot: ... 7 feet 84 twelfths foot; ... 7 feet X 7 twelfths foot foot X 7. = = = .. 4 times $1 × 3 = $3; · · . $1 × 3 = $3 ÷ 4. Or, using q for any quantity, 4 times ... 4 times 19 × 3 = 3q; hence, 19 × 3 = 39 ÷ 4. (4) Or, generally, q× m = mq÷n. For n Such formal proofs are useful and even necessary, but are likely to be misleading unless the pupil has evolved, from rational use of the principle, a clear idea of the relation between times and parts, the importance of which has been emphasized in this book; he is apt to become a mere spectator in the manipulation of symbols, rather than a conscious actor in the mental movement which leads to complete possession of the thought. II. CHANGE OF FORM IN FRACTIONS. 1. From what has been already said, it appears that any quantity may be expressed in the form of a frac tion having any required denominator. Express 9 yards in eighths of a yard. Since the unit of measure is g, 9 such units is 22. Similarly, $7 expressed as hundredths is 188, etc. In general, any quantity of q units of measure expressed as nths is na. n 2. In the same way, any quantity expressed in fractional form may be changed to an equivalent fraction having any denominator. Transform $4 into an equivalent fraction having denominator 20. We can follow either of two plans: (1) 20 is a multiple of 5 by 4; we therefore multiply both terms of the given fraction by 4, getting $18. This is best in practice. (2) Since the new denominator is to be 20, we regard the unit of measure as $88; $t is $ X 4; one fifth of $388; but $ ... $# = = $18. It may be remarked that in such transformations the new denominator is generally a multiple of the original denominator. If it is not, the new equivalent fraction will be complex, it will have a fractional numerator. Thus, if it is required to transform yard to an equivalent fraction with denominator 12, we multiply both 51 terms by 13, with the result 12 3. It is often necessary or convenient to reduce a fraction to its lowest terms-that is, to express it in terms of the largest unit of measure as defined by the unity of reference. This is done by dividing both terms of the fraction by their greatest common measure; thus, $ is equivalent to $4, in which the quantity is expressed in the largest unit of measure, as defined by the unity of reference, the dollar. The principle in 6 volved is that stated in I, 3-viz., the numerical value of the units is increased a number of times, the number of them is diminished the same number of times. In practice, the greatest common measure can generally be found by inspection, as described in Chapter XII. 612 2 X2 X3 X3 X 17 17 In some cases Thus, = = 684 2 × 2 × 3 × 3 × 19 19 the greatest common measure must be found by the general method described in the same chapter. Thus, if the proposed fraction is 79469 265603 we should discover the greatest common divisor to be 13; on dividing both terms of the given fraction there results is the simplest of all equivalent fractions. 6113 20431' which 4. In changing a mixed number to an improper fraction, and vice versa, the primary principle of fractions applies at once: The (1) Reduce 75 yards to an improper fraction. expression 75 yards+yard; express 75 yards in form of a fraction with denominator 3: = 1 yard is 75 yards is yard; yard × 75 = 235 yards; 3 ... 75 yards+yard = 225+2 yards = 227 yards. (2) In the converse operation either consider the problem as a case of formal division giving 75%, or consider the expression as denoting so many thirds of a yard; then 3-thirds one yard; how many 3-thirds in 227 thirds? Evidently, as before, a case of division, giving 75 ones and two-thirds remainder-that is, 75% yards. It may be observed that in (1), while the primary measurement of the quantity is 3 units X 75+2 |