Here Begins Chapter Thirteen on the Method Elchataym and How with It Nearly All Problems of Mathematics Are Solved
Indeed the Arabic elchataym [I] by which the solutions to nearly all problems are found is translated as the method of double false position; one of these problems solved by the method of trees is in the third part of the twelfth chapter; we shall show how to solve this and similar problems in which one need not use the entire elchataym, namely two positions, as these problems can be solved by one of them; and we wish moreover to demonstrate how they and many other problems can be solved by elchataym. Indeed the two false positions are put arbitrarily, when sometimes they both occur smaller than the true argument, sometimes greater, or sometimes one is greater and the other is smaller, and the true argument is found according to the proportion of the difference of one position to the other which is what occurs in the method of four proportionals in which three numbers are known and the fourth unknown, namely the true argument, is found; the first number is the difference between one false position and the other. The second is the difference between the approximations to the true value. The third is the difference which is between the second approximation and the true value. And we will first demonstrate how it is done with the method on hundredweights, so that the three differences are demonstrated subtly with hundredweights, and you will know how to understand the subtle solutions to other problems by elchataym.
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