_x_² _x_₄ cos _x_ = 1 - —————— + —————— - ... 1·2 1·2·3·4 The _tangent_ of _x_ is simply sine of _x_ divided by cosine of _x_. If _y_ is the tangent of _x_, then _x_ is the _antitangent_ of _y_. See also references on trigonometry and on calculus. _Trigonometric_ tables include sine, cosine, tangent, and related functions. Bessel functions These are mathematical functions that were named after Friedrich W. Bessel, a Prussian astronomer who lived from 1784 to 1846. Bessel functions are found as some of the solutions of the differential equation _x_² _Dₓ_(_Dₓy_) + x _Dₓy_ + (_x_² - _n_²)_y_ = O This equation arises in a number of physical problems in the fields of electricity, sound, heat flow, air flow, etc. matrix A _matrix_ is a table (or _array_) of numbers in rows and columns, for which addition, multiplication, etc., with similar tables is specially defined. For example, the matrix ⎮1 2⎮ ⎮ ⎮ ⎮3 4⎮ plus the matrix ⎮5 20⎮ ⎮ ⎮ ⎮60 100⎮ equals the matrix ⎮6 22⎮ ⎮ ⎮ ⎮63 104⎮. (Can you guess the rule defining addition?) Calculations using matrices are useful in physics, engineering, psychology, statistics, etc. To add a _square matrix_ of 100 terms in an array of 10 columns and 10 rows to another such matrix, 100 ordinary additions of numbers are needed. To multiply one such matrix by another, 1000 ordinary multiplications and 900 ordinary additions are needed. See references on matrix algebra and matrix calculus. differences, smoothness, checking On p. 221, a sequence of values of _y_ is shown: 26, 37, 50, 65, 82. Suppose, however, the second value of _y_ was reported as 47 instead of 37. Then the _differences_ of _y_ as we pass down the sequence would not be 11, 13, 15, 17 (which is certainly regular or _smooth_) but 21, 3, 15, 17 (which is certainly not smooth). The second set of differences would strongly suggest a mistake in the reporting of _y_. The _smoothness_ of differences is often a useful check on a sequence of reported values. Supplement 3 REFERENCES A book like the present one can cover only a part of the subject of machines that think. To obtain more information about these machines and other topics to which they are related there are many references that may be consulted. There are still few books directly on the subject of machines that think, but there are many articles and papers, most of them rather specialized.