§ 12-15. NUMERATION. 7

on its left, it stands for ten times as many yards as before. Recollectthis, and you may cease to draw the lines between the columns, be-cause each figure will be sufficiently well known by the place in whichit is; that is, by the number of figures which come upon the right handof it.

13. It is important to recollect that this way of writing numbers,which has become so familiar as to seem the natural method, is notmore natural than any other. For example, we might agree to signifyone ten by the figure of one with an accent, thus, i'; twenty or twotens by 2'; and so on: one hundred or ten tens by j two hundredby 2" ; one thousand by 1 "'; and so on: putting Roman figures foraccents when they become too many to write with convenience. Thefourth number in the table would then be written 2'" 3" 4' 8, whichmight also be expressed by 8 4' 3" 2"', 4' 8 3" 2"'; or the order of thefigures might be changed in any way, because their meaning dependsupon the accents which are attached to them, and not upon the placein which they stand. Hence, a cipher would never be necessary; for104 would be distinguished from 14 by writing for the first i"4, and forthe second F4. The common method is preferred, not because it ismore exact than this, but because it is more simple.

14. The distinction between our method of numeration and thatof the ancients, is in the meaning of each figure depending partly uponthe place in which it stands. Thus, in 44444 each four stands for fourof something; but in the first column on the right it signifies only fourof the pebbles which are counted ; in the second, it means four col-lections of ten pebbles each; in the third, four of one hundred each;and so on.

15. The things measured in (11) were yards of cloth. In this caseone yard of cloth is called the unit. The first figure on the right issaid to be in the units' place , because it only stands for so many unitsas are in the number that is written under it in (6). The secondfigure is said to be in the tens' place, because it stands for a numberof tens of units. The third, fourth, and fifth figures are in the placesof the hundreds , thousands , and tens of thousands , for a similar reason.