line-reckoning, where the lines stood for powers of 10 and the spaces for half of such powers, did not lend itself to this comparison. Accordingly we find such labored explanations as the following, from The Crafte of Nombrynge:
"Euery of these figuris bitokens hym selfe & no more, yf he stonde in the first place of the rewele....
"If it stonde in the secunde place of the rewle, he betokens ten tymes hym selfe, as this figure 2 here 20 tokens ten tyme hym selfe, that is twenty, for he hym selfe betokens tweyne, & ten tymes twene is twenty. And for he stondis on the lyft side & in the secunde place, he betokens ten tyme hym selfe. And so go forth....
"Nil cifra significat sed dat signare sequenti. Expone this verse. A cifre tokens noȝt, bot he makes the figure to betoken that comes after hym more than he shuld & he were away, as thus 10. here the figure of one tokens ten, & yf the cifre were away & no figure byfore hym he schuld token bot one, for than he schuld stonde in the first place...."[[605]]
It would seem that a system that was thus used for dating documents, coins, and monuments, would have been generally adopted much earlier than it was, particularly in those countries north of Italy where it did not come into general use until the sixteenth century. This, however, has been the fate of many inventions, as witness our neglect of logarithms and of contracted processes to-day.
As to Germany, the fifteenth century saw the rise of the new symbolism; the sixteenth century saw it slowly
gain the mastery; the seventeenth century saw it finally conquer the system that for two thousand years had dominated the arithmetic of business. Not a little of the success of the new plan was due to Luther's demand that all learning should go into the vernacular.[[606]]
During the transition period from the Roman to the Arabic numerals, various anomalous forms found place. For example, we have in the fourteenth century cα for 104;[[607]] 1000. 300. 80 et 4 for 1384;[[608]] and in a manuscript of the fifteenth century 12901 for 1291.[[609]] In the same century m. cccc. 8II appears for 1482,[[610]] while MoCCCCo50 (1450) and MCCCCXL6 (1446) are used by Theodoricus Ruffi about the same time.[[611]] To the next century belongs the form 1vojj for 1502. Even in Sfortunati's Nuovo lume[[612]] the use of ordinals is quite confused, the propositions on a single page being numbered "tertia," "4," and "V."
Although not connected with the Arabic numerals in any direct way, the medieval astrological numerals may here be mentioned. These are given by several early writers, but notably by Noviomagus (1539),[[613]] as follows[[614]]:
