"Book of the Abacus," while "Algorithmus Linealis" was applied to numerous works explaining the reckoning on lines which was a slight variation of the abacus idea.
The beautifully printed Calandri arithmetic of 1491, the first in De Morgan's list, differs from its predecessors in having the traditional problems copiously illustrated. The slow-moving snail, who climbs up by day one seventh of a foot and slides back by night one ninth of a foot, is seen here with his head just emerging from the fifty-foot well and looking remarkably active after a climb of 1,575 days. The title page presents Pythagoras as "Pictagoras arithmetice introductor," the wholly erroneous but wide-spread notion being that this philosopher was the originator of the science of numbers.
Of the eight or ten arithmetics (two being parts of compendiums) given by this catalogue as preceding Philippi Calandri's treatise the three following are of general interest. "Prosdocimi de beldamandes algorismi tractatus" (Padua, 1483) contains probably the first reference to a slate; Pietro Borghi, one of these successful text-book writers, wrote the most elaborate of the early books on the subject and more than any other set a standard for the arithmetics of the succeeding century. This text-book for the use of merchants, written in Italian, appeared at Venice in 1484 from the press of Ratdolt. Widman's German text of 1489 employs for the first time the and signs, but simply as warehouse symbols of excess or deficiency.
One of the rarest of the catalogued treasures is the "Arithmetica" or "Compendium of the Abacus" (Turin, 1492) of Francesco Pellos (Pellizzati). It appears that this native of Nice came very near to the invention of decimal fractions, writing almost a hundred years before the first complete explanation of the subject in "La Disme" or "The Decimal" by Simon Stevin the Hollander. Pellos actually used a decimal point to indicate division by such numbers as 100, but its full significance did not dawn on him.
The first printed discussion of arithmetic in the English language is a chapter of Caxton's "The Mirrour of the World or Thymage of the same" (London, 1480); the section begins "And after that of Arsmetrike and whereof it proceedeth." An interesting sidelight is thrown on early American history by the announcement of the discovery in Madrid of the first arithmetic printed in the western hemisphere. Extensive printing was done in Mexico in the second half of the sixteenth century, and it was here that Juan Diaz Freyle published in 1556 the Spanish "Compendium. . . with Certain Rules of Arithmetic."
Of necessity many works apparently unrelated to arithmetic are introduced. The fine distinctions between the sciences did not then exist, so that an astronomer, a geometer, a philosopher or a writer on the Church calendar would not hesitate to bring into his subject a dis-
