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# Benjamin Banneker's Trigonometry Puzzle

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Because the logarithms in the worked examples are all to five decimal places, they do not seem to fit in with any of the above tables.  However, it is instructive to examine the format of Briggs’s logarithms (from Vlacq’s edition of 1628):

 log 26 1, 41497, 33480 log(sin 30) 9, 69897, 00043 log(sin 60) 9, 93753, 06317 log(sin 90) 10, 00000, 00000

Briggs himself defined the first (left-most) comma as standing in for the decimal point; subsequent commas, dividing the figures into groups of five, are merely placeholders to guide the eye.  The logarithms in the worked example can be simply obtained merely by discarding the five right-most digits from the values given in the table.  (As it happens, with the particular numbers chosen by Banneker, no rounding issue arises by the simple truncation of the tabular values.)  It seems highly probable, therefore, that the logarithms used in the example did indeed come from the second (Vlacq) 1628 edition of Briggs’s Arithmetica Logarithmica, or its derivative, the Logarithmicall Arithmetike of 1634 .

At the beginning of the seventeenth century, trigonometric functions were defined as follows:

The radius of the circle was chosen so that the functions could be represented as integers to any desired degree of accuracy.  Napier chose a value of 107(shown here); Briggs chose a value of 1010.  So, for Briggs, the sine of 90° (the whole sine or sinus totus) was 1010  and its (Briggsian) logarithm is, therefore, simply 10...and the logarithms of all the sines in the first quadrant are positive.  In modern trigonometry the radius is taken as unity – a convention unknown much before 1800 – so the sines of the angles in the interval [0°, 90°] are less than unity and their logarithms are, consequently, negative.  The fact that the logarithms of sines in Briggs’s table are positive therefore arises naturally from the way in which the sine function was then defined.

REPRODUCTION OF THE LOG TABLE FROM 1628 courtesy of Brenda Corbin at the United States Naval Observatory, Washington, DC.

Notes:

1.   Banneker’s life can be explored in detail in Silvio Bedini’s Benjamin Banneker, the first African-American Man of Science, Baltimore:  Maryland Historical Society, 1999.

2.   Beatrice Lumpkin’s comments on Banneker’s Trigonometry Puzzle was used for the printed program at a Benjamin Banneker Association session at the NCTM meeting in 1992.  See also her article “From Egypt to Benjamin Banneker:  African Origins of False Position Solutions,” in Vita Mathematica: Historical Research and Integration with Teaching, Ronald Calinger, Ed., Washington:  MAA, 1996, pp. 279-289.  The article is a shortened version of a paper presented at the Quadrennial Meeting of the International Study Group on the Relations Between History and Pedagogy of Mathematics (HPM) at the University of Toronto, August, 1992.