Logarithms: The Early History of a Familiar Function - Before Logarithms: The Computational Demands of the Late Sixteenth Century

The late sixteenth century saw unprecedented development in many scientific fields; notably, observational astronomy, long-distance navigation, and geodesy science, or efforts to measure and represent the earth. These endeavors required much from mathematics. For the most part, their foundation was trigonometry, and trigonometric tables, identities, and related calculation were the subject of intensive enterprise. Typically, trigonometric functions were based on non-unity radii, such as \(R=10,000,000,\) to ensure precise integer output.* Reducing the calculation burden that resulted from dealing with such large numbers for practitioners in these applied disciplines, and with it, the errors that inevitably crept into the results, became a prime objective for mathematicians. As a result, much energy and scholarly effort were directed towards the art of computation.

Accordingly, techniques that could bypass lengthy processes, such as long multiplications or divisions, were explored. Of particular interest were those that replaced these processes with equivalent additions and subtractions. One method originating in the late sixteenth century that was used extensively to save computation was the technique called prosthaphaeresis, a compound constructed from the Greek terms prosthesis (addition) and aphaeresis (subtraction). This relation transformed long multiplications and divisions into additions and subtractions via trigonometric identities, such as: \[2\cos(A)\cos(B)=\cos(A+B)+\cos(A-B).\]

When one needed the product of two numbers \(x\) and \(y,\) for example, trigonometric tables would be consulted to find \(A\) and \(B\) such that: \[x=\cos(A)\;\;{\rm and}\;\;y=\cos(B).\] With \(A\) and \(B\) determined, \(\cos(A+B)\) and \(\cos(A-B)\) could be read from the table and half of the sum taken to find the original product in question. Thus the long multiplication of two numbers could be replaced by table look-up, addition, and halving. Such rules were recognized as early as the beginning of the sixteenth century by Johannes Werner in 1510, but their application specifically for multiplication first appeared in print in 1588 in a work by Nicolai Reymers Ursus (Thoren, 1988). Christopher Clavius extended the methods of prosthaphaeresis, of which examples can be found in his 1593 Astrolabium (Smith, 1959, p. 455).

Finally, with the scientific community focused on developing more powerful computational methods, the desire to capture symbolically essential mathematical ideas behind these developments was also growing. In the fifteenth and sixteenth centuries, mathematicians such as Nicolas Chuquet (c. 1430–1487) and Michael Stifel (c. 1487–1567) turned their attention to the relationship between arithmetic and geometric sequences while working to construct notation to express an exponential relationship. The focus on mathematical symbolism in centuries prior and the growing attention to notation–particularly the experimentation with different versions of exponent notation–played a critical role in the recognition and clarification of such a relationship. Now the mathematical connection between a geometric and an arithmetic sequence could be made all the more apparent by symbolically capturing these sequences as successive exponential powers of a given number and the exponents themselves, respectively (see Figure 6). The work on the relationships between sequences was mathematically important per se, but was equally significant for providing the inspiration for the development of the logarithmic relation.

________________________

*Note: Modern trigonometry is essentially based on triangles inscribed in a unit circle; that is, a circle with radius \(R = 1\). Early practitioners used circles with various values for the radius. The relationship between the modern sine function and a sine or half-chord in a circle of radius \(R\) is given by \({\rm Sin}\,{\theta}=R\sin \theta\), where the modern sine function has a lower case '\(\rm{s}\)' and the pre-modern sine an upper case '\(\rm{S}\)'.