UNIT or UNITY, the name given to that magnitude which is to be considered or reckoned as one, when other magnitudes of the same kind are to be measured. It is not itself one, but is the magnitude which one or 1 shall stand for in calculation: it is a length, or a weight, or'a time, as the case may be, while 1 is only a numerical symbol. This symbol 1 represents the abstract conception of singleness, as distin guished from multitude, and is the unit of abstract arithmetic; but all concrete quantities must have units of their own kind.
Unity, says Euclid (book vii, def. 1), is that according to which each of existing things is called one : Memoir ?art, KO guaarov Ta?P 6prcey ev M-yerat. And, allowing somewhat for idiom, it would not be easy to mend this definition. Anything may be unity, fir things of jts own kind.
The common division of units into abstract and concrete is merely the distinction between the unit of numeration and that of measure ment: the former implying that reckoning or computation is to be performed, without specific reference to any particular object of reckoning; the latter, that some certain unit of length, of capacity, or whatever it may be, is to bo signified by 1, On this point the learner must take pains to see, that of all the fundamental operations of arithmetic, three are wholly independent of this distinction, which cannot be said of the fourth. Addition, subtraction, and division can bo physically performed, and without reference to units : two lines may be put together into one line, a line may be cut off from another, or a line may be carried along another time after time, until it is seen how many times the greater contains the less. But multiplication requires that number or magnitude should be taken a number of times, and the Idea of multiplying a magnitude by a magnitude involves an absurdity. [MULTIPLICATION; RECTANGLE.] Nevertheless some enter
prising writera on arithmetic profess to multiply magnitude by magni tude; and, to make their doings more striking, they often choose for their instance to multiply 99/. 19s. 11 id. by 991. 19s. Hid. To take a humbler case, let us examine the product of 5 shillings and 3 ahillings : beginners educated in the common system of.arithmetic are generally loth to part with the idea that must be 15 shillings.
The common rule of three, as generally stated, and given without proof, is the cause of much of the habit which leads to this unwilling. new; and for those who cannot see any difference between 5 shillings taken 3 times, and 5 shillings multiplied by 3 shillings, the examination of a question in this rule will be worth while. Let it be as follows : If 10 apples cost 7 pence, how much will 30 apples cost/ The computer proceeds in this manner :—As 10 apples are to 7 pence, so are 30 apples to the answer required. According to the rule, he multiplies together 7 pence and 30 apples, and produces 210—of what ho does not say. They can hardly be simple apples, or pence: probably they are 210 chemical compounds of an apple and a penny. The result is to be divided by the first term, 10 apples: here 210 divided by 70 gives 21, and the apples in the divisor decompose this compound, free it of all its fruit,and leave for the final answer 2] pence. The confusion which is caused by the improper use of the concrete unit can hardly be conceived by any but one who has been used to teaching.