# Briggs' TRIGONOMETRIA BRITANNICA

## translated and annotated byIan Bruce

Briggs' original book has no table of contents. The first 17 chapters comprise Book I, written by Briggs before his death in 1631, are concerned mainly with the construction of Tables of Sines. The work was completed by his friend and associate Henry Gillibrand, and published in 1633 by Adrian Vlacq in Gouda.

CONTENTS

To the Cultivator of Mathematics : An obituary on Briggs by Gellibrand. Link to this document by clicking here. You can do likewise to access any chapter; use the browser 'Back' arrow to return to this screen.

Part I

Chapter one
Briggs makes some prefatory comments here regarding a decimal form of measuring angles and fractional parts of angles that had been suggested earlier by Vieta, and sets out the number of places in his tables; he does not, however, use the decimal point.

Chapter Two
Some Lemmas on the original method of Ptolemy for calculating a table of sines are briefly presented.

Chapter Three
In which a general method for the triplicating of an arc is presented.

Chapter Four
The trisection of arcs, or the finding of cube roots, is considered.

Chapter Five
Five-fold multiplication of an arc is considered. This may extend the original arc beyond a whole circle.

Chapter Six
The inverse process of finding the fifth part of an arc, or the 5th root, is here considered.

Chapter Seven
Division of an arc into 7 equal parts; or finding the 7th root.

Chapter Eight
Introduction to a general method for finding the coefficients in the expansion for a multiple angle in terms of a part: The 'Abacus Panchrestos'; predating Pascal's Triangle. .

Chapter Nine
Examples of the use of the above table, and the extraction of even roots.

Chapter Ten
The geometrical justification of some of the procedures introduced above.

Chapter eleven
The derivation of sectional equations is presented in an alternate manner that originated with Vieta.

Chapter twelve
The method of subtabulation used extensively by Briggs is introduced. Even and odd finite differences are discussed.

Chapter thirteen
The mechanics of setting up a table of reference sines gets under way at last.

Chapter fourteen
The scheme for dividing the circle into 100 equal parts, and decimal fracions of degrees.

Chapter fifteen
Tables for tangents and secants.

Chapter sixteen
Logarithms of sines.

Chapter seventeen
Logarithms of tangents and secants and many theorems that are useful in their evaluation. etc.

PART II

Plane Triangles :
Right angled triangles and subsequently all kinds of plane triangles are solved, using a combination of rules, including half angle tangent formulas.

Quadrantal Triangles Part I :  This is a useful introduction to the art of spherical triangles, taken from a purely geometric viewpoint, rather than the applied approach of the navigator or astronomer. A number of  the relevant propositions are derived from two axioms, while Napier's rules of circular parts are finally introduced and a number of right angled spherical triangles are solved. In so doing, a complete description of the initial triangle is added in a diagram including completed quadrants for each problem, in which the terms used in the ratios are shown : the proportions used can always be found from Napier's rules, and also from Gellibrand's own rules, which relate to earlier works, such as those of Pitiscus, etc.

Quadrantal Triangles Part II :  This completes the treatment of spherical triangles, where oblique angled triangles are separated into two right angled triangles, which are then treated separately as in Part I. This leads to more complicated diagrams, but still understandable. The brief paper of Euler mentioned is worth looking at to see some of the results used in a more modern setting.

Ian Bruce. August 2006/February 25th, 2013.  latest revision. Copyright : I reserve the right to publish this translated work in book form. However, if you are a student, teacher, or just someone with an interest, you can copy part or all of the work for personal or educational use.