FULLER’S CALCULATING RULE.—This instrument, which is shown in Fig. 18, consists of a cylinder _d_ capable of being moved up and down and around the cylindrical stock _f_, which is held by the handle. The logarithmic scale-line is arranged in the form of a helix upon the surface of the cylinder _d_, and as it is equivalent to a straight scale of 500 inches, or 41 ft. 8 in., it is possible to obtain four, and frequently five, figures in a result. Upon reference to the figure it will be seen that three indices are employed. Of these, that lettered _b_ is fixed to the handle; while two others, _c_ and _a_ (whose distance apart is equal to the axial length of the complete helix), are fixed to the innermost cylinder _g_. This latter cylinder slides telescopically in the stock _f_, enabling the indices to be placed in any required position relatively to _d_. Two other scales are provided, one (_m_) at the upper end of the cylinder _d_, and the other (_n_) on the movable index. [Illustration: FIG. 18.] In using the instrument a given number on _d_ is set to the fixed index _b_, and either _a_ or _c_ is brought to another number on the scale. This establishes a ratio, and if the cylinder is now moved so as to bring any number to _b_, the fourth term of the proportion will be found under _a_ or _c_. Of course, in multiplication, one factor is brought to _b_, and _a_ or _c_ brought to 100. The other factor is then brought to _a_ or _c_, and the result read off under _b_. Problems involving continuous multiplication, or combined multiplication and division, are very readily dealt with. Thus, calling the fixed index F, the upper movable index A, and the lower movable index B, we have for _a_ × _b_ × _c_:—Bring _a_ to F; A to 100; _b_ to A or B; A to 100; _c_ to A or B and read the product at F. The maximum number of figures in a product is the sum of the number of figures in the factors and this results when all the factors except the first have to be brought to B. Each time a factor is brought to A, 1 is to be deducted from that sum. For division, as _a_/(_m_ × _n_), bring _a_ to F; A or B to _m_; 100 to A; A or B to _a_; 100 to A and read the quotient at F. [Illustration: FIG. 19.] The maximum number of figures in the quotient is the difference between the sum of the number of figures in the numerator factors and those of the denominator factors, _plus_ 1 for each factor of the denominator and this results when A has to be set to all the factors of the denominator and all the factors of the numerator except the first brought to B. Each time B is set to a denominator factor or a numerator factor is brought to A, 1 is to be deducted. Logarithms of numbers are obtained by using the scales _m_ and _n_ and hence powers and roots of any magnitude may be obtained by the procedure already fully explained. The instrument illustrated is made by Messrs. W. F. Stanley & Co., Limited, London.