Patent GB-786765-A (GB)
Title: Mechanism for solving the cubic equation with three real roots
Inventor: SODDY FREDERICK
IPC: G06G3/00
Filed: 1955-03-30
Abstract: 786,765. Calculating apparatus. NATIONAL RESEARCH DEVELOPMENT CORPORATION. March 29, 1956 [March 30, 1955], No. 9227/55. Class 106 (1). Apparatus for finding the three real roots of cubic equations in the formula:- is based on properties of the geometrical figure known as the arbelos which consists, Fig. 1, of two semicircles enclosed by a third, and with their centres A, B and C colinear. It can be shown that, if the radius of the P circle contacting these semicircles is constant and equal to unity, the roots of the above equation, for all values of f, are given by the radii of the three semicircles. Two mechanical systems are described for establishing the arbelo construction of unit radius P circle, whilst adjusting it to different f values, so that the corresponding roots can be read off. The first, Figs. 2 and 4, makes use of a system of cords and pulleys which, for convenience, is duplicated symmetrically with respect to the line ABC. An upright board with a vertical guide channel accommodates blocks representing the points A, B and C, and pulleys P and P2 for the point P are spaced two units distance from the channel. Three cords attached to a weight pass over each of the pulleys P1 and P2 and are taken to the block C, one directly and the other two round pulleys on the blocks A and B. In this manner the lengths PB+BC, PA+AC and PC vary equally as the blocks are moved and, by this means, the arbelo relationship of the points A, B and C is maintained. After adjusting the apparatus, e.g. by means of a screw applied to the block A, two of the roots are read on a scale, carried by the block A, against index marks on the blocks B and C. The third root is the difference of these two. Two alternative methods are adopted for setting the mechanism to given values of f, these setting mechanisms being disposed on the reverse side, Fig. 4, of the board. The first makes use of the fact that the line AF, Fig. 1, drawn from A to the apex of an equilateral triangle on BC, is f units in length. The equilateral triangle is formed by celluloid strips, shown by broken lines, secured to the blocks B and C and a scale of f values is pivoted to the block A. The mechanism is adjusted until the apex of the equilateral triangle coincides with the given f value on the scale. The second method makes use of the fact that the sum S of the distances moved by the three blocks =#(4f-27). A cord, stretched between fixed points L and J, is taken round pulleys on the blocks B and C and a block D moving with A, and also round a pulley on a block E adapted to travel along a scale of S values. The second system described employs positive links in place of cords and uses the points C, V and S, Fig. 10, of the arbelos instead of the points A, B and C, Fig. 1. The P circle radius is again kept constant at one unit, so that PQ= PX=XD=1, the angle VQS being 45 degrees. The apparatus, Fig. 12a, comprises a horizontal board 1 supporting rails 3 on which slide three swivel blocks C, V and S, corresponding to these points of Fig. 10. Points P and X are represented by blocks pivoted to the board and distant 2 units and 1 unit from the CVS line. An operating rod carried by block P slides through block C and pivotally carries at Q, distant one unit from P, two further rods set at 45 degrees and slidable in the block S and in the blocks X and V respectively. This construction maintains the arbelo relationship of the blocks C, V and S as the operating lever is moved. In Fig. 10, the sum of the movement of C with half that of V is equal to S/2, S being (as above) #(4f-27). Therefore, to adjust the apparatus, an index 5 on block C co-operates with an S-scale 12, which is driven oppositely to the block V and at half the rate by means of reduction gears on a shaft 9 meshing with racks on the scale and block V. The two roots are shown by index pointers 6 and 7 on the blocks C and V against graduations on a scale 4 carried by the block S.