3iH Colorado College Studies. 



"Measuring in the ordinary sense — the process which leads to the 

 representation of continuous magnitudes as lines or surfaces, in terms 

 of some unit of measure — (.leserves all the prominence which our au- 

 thors [McLellan and Dewey] would give it in arithmetic. We do not 

 mean measuring in the exact mathematical sense, of course, but the 

 rough measuring of common life, in which the magnitude measured and 

 the unit are always assumed to be commensurable. Compared with 

 counting, or even addition and multiplication, an operation which in 

 volves the use of an arbitrary unit, and the comparison of magnitudes 

 by its aid, is artificial. But this metrical u.se of numVier is of immense 

 practical importance, and of great interest to any child mature enough 

 to understand it. No doubt a child may use a twelve inch rule to ad- 

 vantage when practicing multiplication and division of integers. Cer- 

 tainly, such an aid is almost indispensable in learning fractions. Without 

 it, the fraction is more likely to be a mere symbol to him, without exact 

 meaning of any kind. ' Two-thirds ' has a reality for the child who can 

 interpret it as the measure of a line two inches long in terms of a unit 

 three inches long, which it quite lacks for him who can only repeat that 

 it is 'two times the third part of unity.' Mathematicians now define 

 the fraction as the symbolic result of a division which cannot be actually 

 effected, but that definition will not serve the purposes of elementary 

 instruction. It is as certain that the fraction had a metrical origin as 

 it is that the integer had not, and in learning fractions, as in learning 

 integers, the child cannot do better than follow the experience of the 

 race."'— Pkof. H. B. Fine, in Science. N. S. Vol. Ill, 1890, p. 136. 



The mo.st ancient mathematical hand-book known to our 

 time — the Ahmes papyrus, about 2000 B. C, whii-h claims to 

 1)6 founded on much older Egyptian documents — begins with 

 fractions. It was probably written for the advanced mathe- 

 maticians of its day. The study of this document shows how 

 difficult fractions were to the ancients. Ahmes confines him- 

 self to vnit-fraclions having unity for their numerators. 

 If another fractional value was to be considered, it was al- 

 ways expressed as the sum of two or more unit-fractions. 

 Thus: 2*i = i'?+4'a. And how did the necessity of the in- 

 troduction of fractions arise? The document contains prob- 

 lems like this, " Divide 2 by 3," and Ahmes solves this by 

 means of his fractions. Thus dividing 5 by 21 gives \ \h h. 

 That the idea of measurement Avas predominant in the use 

 of these fractions follows from such problems in Ahmes as 



