Ian Bruce

University of Adelaide, Australia

[*Briggs' original book has no table of contents. Most chapters have added explanatory Notes by the translator that may include references; otherwise, the subject matter should be intelligible as it stands. Although some attempt has been made to fit the work into the contemporary mathematical scene, and Hutton *(*in the preamble to his Mathematical Tables, p. *75)*, asserts that Briggs' methods are peculiarly his own, nevertheless it is felt that the underlying influence of Viete is never far away.*]

A Valediction by Henry Gellibrand

Chapter 1: The definitions of the sine, tangent, and secant ratios. Briggs indicates his desire to replace conventional degrees and their sexagesimal fractions with decimal degrees. The great contribution of Ptolemy and later workers acknowledged. Notes included.

Chapter 2: *Concerning How the Subtended Chords Are Sought in the Writings of the Ancients*. The calculation of the lengths of sides of regular figures inscribed in a circle. Illustration of the use of a Theorem by Ptolemy to find an unknown length of an inscribed quadrilateral. Notes.

Chapter 3: To triplicate or trisect a given arc. A cubic equation relating a chord subtending a given angle to the chord of triple the angle is derived geometrically for the acute, obtuse, and reflex cases of triplication. Notes.

Chapter 4: Concerned with the solution of the cubic equation for the chord length of the trisected arc for various sizes of angles. This is done numerically using a method identical to the Newton-Raphson procedure. Notes.

Chapter 5: To quintuplicate or quinquisect a given arc. A quintic equation relating the chord subtending a given angle to the chord of the quintuple angle is examined for various cases for quintuplication. Notes.

Chapter 6: Concerned with solving the quintic equation for the chord length for the quinquisection of an arc, taking the chord for 72 and related arcs. Again, this is done numerically using an extension of the above method for four of the roots, identical to the Newton-Raphson procedure. Extensive Notes.

Chapter 7: To septisect a given arc. A 7^{th} power equation relating the chord subtending a given angle to the chord of the septuple angle is examined. Notes.

Chapter 8: A general method is established for establishing the coefficients of the powers for the equation of any odd section. A table of binomial coefficients is formed, the *Abacus Panchrestus*, and the coefficients are found from a related Second Table.

Even sections of arcs are considered that give rise to even power equations for the squares of lengths of chords: the coefficients again being found from the Second Table. The square of the length of the chord of an even multiple of the basic arc can be found from the various even powers of the basic squared chord. Briggs performs a test using 12 and 72 arcs to vindicate his method. Extensive Notes.

Chapter 9: The quadratic equation for the bisection of a chord is solved by the 'algebraic' (i.e. numerical) procedure, and the common quadratic formula is also used to find the roots, as a further test. Further, a trisection is performed on the squared chord of 72, as further evidence for the validity of the method. Notes.

Chapter 10: Two useful Theorems are given: 1. Relates squared chords to other chords of known angles; and 2. Relates known chords to other squared chords of known angles.

Chapter 11: Two Theorems are given that show how to find the lengths of the various chords from a common vertex in a regular inscribed figure, in a alternative scheme that uses all the powers of the basic subtended chord, which are tested on the pentagon and 20-gon. Notes.

Chapter 12: A basic Table of Sines at 3 ^{1}/_{8} spacing is presented. A difference table is produced. It is shown that the differences of the Sines of equal arcs are proportional to the sines of the complements of the mean arcs. Odd and even differences of higher order can then be computed separately from proportionality.

The interpolation scheme used in the *Arithmetica Logarithmica* is introduced, and the sines at intervals of ^{1}/_{5} of the original spacing are found from the corrected differences.
The 6^{th} powers of the whole numbers between 50 and 70 are found by using the interpolation scheme, from the given 50^{6}, 55^{6}, 60^{6}, 65^{6}, 70^{6}.
Extensive Notes.

Chapter 13: The route by which the above Table of Sines has been found is set out, which is extended ^{5}/_{8}^{th} for the quadrant by interpolation. The subtabulation is continued by repeated division by 5 to show how the sines of angles at ^{1}/_{1000}^{th} degree spacing can be achieved.

Chapter 14: A basic table of 25 values, setting out the alternative sines where the new degree is ^{1}/_{100}^{th} part of the circumference, and each degree to be divided into 1000 equal parts. Note.

Chapter 15: Theorems are presented by means of which the tangents and secants of angles can be found from the extended Table of 144 Sines in the quadrant..

Chapter 16: Concerning mainly the Logarithms of Sines. These are produced by proportionality from a few known values; the characteristic will be 9 for angles above 5:44'. Note.

Chapter 17: Concerning Tangents and Secants and their Logarithms: a number of miscellaneous theorems in a continuation of Ch. 15.

Ian Bruce January 2003

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