- If terms of the same order occur more than once in an equation, then they are combined, if necessary by addition or subtraction on both sides of the equation.
- If there are terms in an equation that are preceded by a minus sign, they are equalized by addition.
- If root terms occur, then square.
- If there are fractions on both sides, then multiply “cross-white”.
The third section of the second book contains over 400 problems with complete solutions on 145 double pages (determining the variables, setting up the equation, transformations according to the rules given above). Apart from the high amount of verbal content, the solution methods hardly differ from those of the algebra problems as we know them from today’s school books. Here is a selection of the first type:
- If you multiply a number less than 10 by 3, the product is seven times as much over 10 as the number less than 10.
- Of nine numbers in an arithmetic sequence (“increasing over itself with the same transgression”), the smallest is the number 4 and the sum is 48.
- Find the length of an altitude in a triangle with sides 13, 14, 15.
- In a right-angled triangle (»right angular triangle«), one of the legs has the length 3 + √18, the other leg and the hypotenuse together have the length 9 + √162.
The total of 240 problems on equations of the first type deal with problems that already appear in Leonardo of Pisa’s collection of problems and that can be found later in Leonhard Euler’s “Complete Guide to Algebra”.
It’s about gamblers winning or losing different amounts, about travelers moving towards each other from two cities. Then two men compare the contents of their purses; three men would like to buy a house or a whole village or a horse or several horses, but they need money from the others to do so. Next follow calculations for the salary of lansquenets to conquer a city. Workers are remunerated for working days and money is deducted for missed days, you have to break down the income from a bridge toll afterwards. Grain is ground simultaneously and for the same length of time in three mills with different capacities; corporations are formed and profits shared; in another story, a musician visits three taverns in succession and is rewarded for doing so, but has to pay for his food each time, and so on.
In the case of equation systems with two or even more than two variables, one needs an additional variable, which Rudolff calls “quantity”. The solution to these tasks (from today’s point of view) still seems somewhat complicated, but only a few years later Stifel already found a solution variant with several variables. Among the tasks there is also a problem with a negative solution, which Rudolff comments with “folgt immuglichkeit”. Overall, the high proportion of tasks in which root terms occur is striking; this also applies to the other equation types, for example: