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 a problem which he could not do without counters. Iago (in Othello, i. 1) expresses his contempt for, "forsooth a great mathematician," by calling him a "counter-caster." So general, indeed, says Peacock, appears to have been the practice of this species of arithmetic, that its rules and principles form an essential part of the arithmetical treatises of that day. The real fact seems to be that the old methods were used long after the Hindoo numerals were in common and general use. With such dogged persistency does man cling to the old!

The Liber Abaci was, for centuries, the storehouse from which authors got material for works on arithmetic and algebra. In it are set forth the most perfect methods of calculation with integers and fractions, known at that time; the square and cube root are explained; equations of the first and second degree leading to problems, either determinate or indeterminate, are solved by the methods of 'single' or 'double position,' and also by real algebra. The book contains a large number of problems. The following was proposed to Leonardo of Pisa by a magister in Constantinople, as a difficult problem: If A gets from B 7 denare, then A's sum is five-fold B's; if B gets from A 5 denare, then B's sum is seven-fold A's. How much has each? The Liber Abaci contains another problem, which is of historical interest, because it was given with some variations by Ahmes, 3000 years earlier: 7 old women go to Rome; each woman has 7 mules, each mule carries 7 sacks, each sack contains 7 loaves, with each loaf are 7 knives, each knife is put up in 7 sheaths. What is the sum total of all named? Ans. 137,256.[3]

In 1220, Leonardo of Pisa published his Practica Geometriœ, which contains all the knowledge of geometry and trigonometry transmitted to him. The writings of Euclid and of some other Greek masters were known to him, either from Arabic