TFiE NuMBEU Concept. 39 



the computation of the average daily produce of fat, when 

 the yearly produce is ten "besha," and from the geometrical 

 examples dealing with areas of land. For instance, a trian- 

 gular piece of land is computed as having half the area of 

 the corresponding parallelogram. 



Apparently, number tirst suggests itself to the child in 

 connection with discrete things. He sees three distinct ob- 

 jects, and learns to count one, fivOj three. In measurement 

 (of lengths, for instance) the mind marks otf some unit of 

 length (the foot) along the length to be measured (a yard- 

 stick, let us suppose), and thereby conies to imagine the 

 yard as made up of parts; the measured length is thought of 

 as composed of three equal parts; that is, by a mental act the 

 continuous length is pictured as made up of discrete parts of 

 a group of like objects. The original or primitive yard-stick 

 is differentiated by the mind into an artificial group of three 

 equal lengths. A pencil is found to be six inches long; the 

 mind at once pictures a group of six equal lengths, which 

 have become discrete objects of thought. In the counting of 

 separate objects, we, by a process of abstraction, consider the 

 objects as alike; in measuring, by a further mental process, 

 we consider a continuous magnitude as made up of separate 

 like parts.* 



In measurement we extend the number concept so that it 

 is applicable not only to things that are discrete or discon- 

 tinuous, but also to things which are continuous. AVe can 

 now say that ratio is a nnm1>er, but we are not allowed to say 

 that number is alwdys rcitio. Number in general is a broader 

 term than ratio. 



While in the crude measurements of every day life all 

 magnitudes appear to us as commensurable with one another, 

 mathematical reasoning shows that incommensurability may 

 exist. The keen intellect of the Greeks tirst detected the 

 fact that the side of a square and its diagonal are incommen- 

 surable with each other. Hence in the refined reasoning of 

 the mathematician it is not sutiicieut for measurement to 

 deal with ratios which are integers or ordinary fractions, but 

 with numbers which are incommensurable to the measuring 



♦Consult an article by Prof. G. B. Halsted, in .Vcjeuce, N. S., Vol. Ill, pp. 170, 

 471, to which I am iudubted. Bee also Clifford, loc. cit., p 95. 



