So, too, in Chaucer's explanation of the astrolabe,[478] written for his son Lewis, the number of degrees is expressed on the instrument in Hindu-Arabic numerals: "Over the whiche degrees ther ben noumbres of augrim, that devyden thilke same degrees fro fyve to fyve," and "... the nombres ... ben writen in augrim," meaning in the way of the algorism. Thomas Usk about 1387 writes:[479] "a sypher in augrim have no might in signification of it-selve, yet he yeveth power in signification to other." So slow and so painful is the assimilation of new ideas. Bernelinus[480] states that the abacus is a well-polished board (or table), which is covered with blue sand and used by geometers in drawing geometrical figures. We have previously mentioned the fact that the Hindus also performed mathematical computations in the sand, although there is no evidence to show that they had any column abacus.[481] For the purposes of computation, Bernelinus continues, the board is divided into thirty vertical columns, three of which are reserved for fractions. Beginning with the units columns, each set of {122} three columns (_lineae_ is the word which Bernelinus uses) is grouped together by a semicircular arc placed above them, while a smaller arc is placed over the units column and another joins the tens and hundreds columns. Thus arose the designation _arcus pictagore_[482] or sometimes simply _arcus_.[483] The operations of addition, subtraction, and multiplication upon this form of the abacus required little explanation, although they were rather extensively treated, especially the multiplication of different orders of numbers. But the operation of division was effected with some difficulty. For the explanation of the method of division by the use of the complementary difference,[484] long the stumbling-block in the way of the medieval arithmetician, the reader is referred to works on the history of mathematics[485] and to works relating particularly to the abacus.[486] Among the writers on the subject may be mentioned Abbo[487] of Fleury (c. 970), Heriger[488] of Lobbes or Laubach {123} (c. 950-1007), and Hermannus Contractus[489] (1013-1054), all of whom employed only the Roman numerals. Similarly Adelhard of Bath (c. 1130), in his work _Regulae Abaci_,[490] gives no reference to the new numerals, although it is certain that he knew them. Other writers on the abacus who used some form of Hindu numerals were Gerland[491] (first half of twelfth century) and Turchill[492] (c. 1200). For the forms used at this period the reader is referred to the plate on page 88.