 Most of the 600 pages of labor Abakai consists of word problems solved using a variety of methods. Leonardo uses the rule of three as well as false position. Most of these problems give more insight into medieval culture than mathematics. For example, a man buys eggs at seven eggs for a denaro and sells them at five eggs for a denaro. If his profit is 19 denarii, how much did he invest in eggs? Leonardo supposes the man buys 35 eggs, presumably because that's a common multiple of seven and five. Since he bought 35 eggs at seven eggs for a denaro, he spent five denarii. Since he sold 35 eggs at five eggs for a denaro, his revenue was seven denarii. So for an investment of five denarii, he earned a profit of two denarii. So earning a profit of 19 denarii would require an investment of five times 19 halves, 47 and a half denarii. Leonardo also solves several vat problems. We might call them work problems today. So here we have a vat with four outlets. The first can empty the vat in one day, the second in two, the third in three, and the fourth in four. How long will it take to empty the vat if all four outlets are used? So Leonardo assumes the outlets are kept open for 12 days, again presumably because 12 is a common multiple of one, two, three, and four. The first outlet would empty the vat 12 times, the second six times, the third four times, and the fourth three times. So the four outlets working together would empty the vat 25 times in 12 days, so the vat would empty one time in 12 25ths of a day. Leonardo also finds the number of hours by multiplying by 12 to get five and 19 25ths hours. In the 13th chapter, Leonardo introduced what he called el-shat time. We call this the method of double-false position or linear interpolation. As you might guess from the name, it's something Leonardo picked up from Arabic sources. We call it the method of double-false position or linear interpolation. Begin with two guesses for the solution and use proportional reasoning to solve the problem. So in another alloy problem, one silver alloy has three ounces of silver per pound and the second has six ounces of silver per pound. How many pounds of each alloy? I need to produce 15 pounds of an alloy with five ounces of silver per pound. Leonardo begins by noting 15 pounds of an alloy with five ounces of silver per pound requires getting 75 ounces of silver. For an initial guess, Leonardo supposed we took three pounds of the first alloy. This gives nine ounces of silver. To make a total of 15 pounds, another 12 pounds of the second alloy are used, which gives 72 more ounces of silver. Together, these produce 81 ounces of silver, which is six ounces too many. Next, Leonardo assumed we took four pounds of the first alloy, which gives 12 ounces of silver. Since we want 15 pounds total, we take 11 pounds of the second alloy, giving 66 ounces of silver, or 78 ounces altogether, which is now three ounces too many. So to summarize what we have so far, three pounds of the first and 12 pounds of the second give six ounces too many, four of the first and 11 of the second give three ounces too many. And note that increasing the amount of the first alloy by one pound decrease the amount of silver by three ounces. Since we're currently three ounces over the amount we want, we should increase the amount of the first alloy by one more pound. So the solution is five pounds of the first alloy and ten pounds of the second.