Tip 74. Two operators of input and solution of a system of the algebraic equations

Fig. 74. Nautilus I

The attempts of solving the problem about the sizes of the submarine Nautilus from the famous roman of Jule Vern Twenty thousand league[1] under water and Mysterious island are made in the fig. 74. If we take description of the submarine without assumtions and reservations[2] (see the beginning of the fig. 74) then the problem is reduced to solution of two nonlinear algebraic equations with three unknowns. The solution of such (underdetermined[3]) problem will be not simply three number (dimension of the submarine: diameter of the cylinder d, length of the submarine L and length of its crown Lc), but the dependence of two parameters of the submarine for example the dependence of its length and diameter from its third parameter. Three points of such dependence are obtained in the fig. 74.

Main point of the tip: when you solve the system of the algebraic equations numerically then the values of all variables are given. At that one part of the variables is constants, the other part is unknowns of the system, which values are selected by the function Find such way that the equations would become the identities. It is worth to write this feature by corresponding not before the key word Given: the values of the constants are given by the operator º, but the values of the variables by the operator :=. This way was dome in the fig. 74 in three attempts (methods) of calculation of the sizes of the Nautilus. When the constant that is given by user by turns becomes one of the variables: d (diameter of the submarine the first solution), L (its length the second solution) and Lc (length of its bit core the third solution).

Our three solutions in the fig. 74 are closed in the limits of the areas (area), that are ready for protection from editing (Lock) and/or for slam (Collapse) see tip 62. Owing to it the solutions in the fig. 74 can be overlooked not all at once but only by turns on a display.

[1] From Popular illustrated dictionary of history and geography published in France last century in the article about league (une lieue) it is possible to know that in X century one league equaled 1 500 Roman steps (2 222 m). In XVIII century post league was 3 894 m, but general league was 4 445 m. The length of league changes in time and space. In the same XVIII century the league was 3 898 m in Turin, in Bern the league was 4 278 m, in Burgundy it was 5 169 m, in Provence and Gascon the league was 5 847 m and etc. In the second part of XIX century (when J. Vern had been writing its roman) the sea league equaled 5592 in France and England, the astronomical one equaled 6418 m, but the metrical league (simple average of all league) equaled 4 km. That why followed the fuzzy-set theory the title of this roman we should read the following way: Very many leagues under water. But this note could be considered as the continuation of the sketch Mathcad and some secrets of literature (http://twt.mpei.ac.ru/ochkov/Gerasim/Gerasim.htm).

It is possible to dispute not only about the length of the league but also about gender of this noun. In French the league is female une lieue. In our note Russian word the league is neuter but the author is not sure in it. It is said that widely officer rank lieutenant comes from this word the league kept a place (span (league) of own ground).

[2] But assumtions and reservations may be the following: the submarine is not round straight cylinder with two round right cones along the edges, but it is only something reminiscent this composite solid; the section of the submarine does not have to be round necessarily; it is not said exactly what area is meant: surface area of the submarine or, one might say, living area, area of inside rooms and etc.

[3] Underdetermined problem we have in the sketch Mathcad and some secrets of literature (http://twt.mpei.ac.ru/ochkov/Gerasim/Gerasim.htm): when we calculated rubles price it was possible to use not seven but three algebraic equations. In underdetermined problem about the size of Nautilus we have two equations but we need three ones as since there are three unknowns.