Ian 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 25^{th},
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.