# Briggs' ARITHMETICA LOGARITHMICA

## translated and annotated byIan Bruce University of Adelaide, Australia

Acknowledgments:
I wish to express my gratitude to the managers of the MacTutor website, and JOC in particular, for allowing me to display my work here. To those who have helped me in some way with this project, I say a big: Thank You.

It is hoped that anyone with in interest in the history of mathematics, be they student or teacher, will gain from reading at least some of these chapters. Perhaps the best way to enter the spirit of the thing is to ask: How did people reason mathematically before the invention of algebra?

The links to the chapters below lead to pages in PDF format.

## CONTENTS

[Briggs' original book has no table of contents. The first 14 chapters are concerned with the construction of Logarithms. I have used some of Hutton's wording from p.61 - p.75 of the preamble to his Tables to introduce the chapters.]

Some biographical notes
A mainly mathematical biography of Briggs, relying mainly on J. Ward's : Lives of the Professors of Gresham College. Biographical material is also inserted in the various chapter notes and comments as appropriate

General notes on the Translation
Some prefatory comments are made in this chapter concerning the sectional layout adopted in the translation: Each chapter has four sections: - The first section is a chapter synopsis ; the second the translation; the third notes and comments ; and the fourth the Latin original.

Chapter one
The origin of Logarithms as a correspondence between two sets of numbers is set out; the first set being a geometric progression; the second set an associated arithmetic progression. Arbitrary series of numbers as well as the base ten Logarithms are examined initially, and two axioms established for further use.

Chapter two
The convenient choice of zero as the Logarithm of one is made; following from this, the Logarithms of other whole numbers are either the indices of powers of ten, or are numbers proportional to them. The log of a product as the sum of the logs of the factors and the log of a quotient as their difference is established.

Chapter three
Rational Logarithms as indices of powers of ten and by the taking of roots of 10.

Chapter four
The characteristic.

Chapter five
Base ten Logarithms can be found by two methods: one due to Napier, which is expanded on at length to find log 2 and log 7, essentially by counting the number of figures in very large equal powers of 2 and 10; while the other method is the main subject of this book.

Chapter six
The Logarithms are formed from continued means: in which the repeated square root of 10 is taken to establish eventually a proportionality between the fractional part of the root and the index. Some non-fatal but time wasting errors are uncovered by the translator.

Chapter seven
The Logarithm of 2 is found by the Radix Method, and subsequently the logs of 5 and 3.

Chapter eight
The continued extraction of square roots of numbers just larger than one is facilitated a method invented by Briggs relying on finite differences.

Chapter nine
An ingenious method is found by Briggs to find the logs of prime numbers.

Chapter ten
The Logarithms of fractions is considered.

Chapter eleven
Use is made of proportional parts to increase the accuracy of Logarithms found in the Chiliades.

Chapter twelve
The first method of subtabulation, used extensively by Briggs.

Chapter thirteen
The second method of subtabulation, proposed by Briggs for the completion of the tables, relying on central finite differences of orders up to 20. The most ambitious mathematics in the book, but not explained by Briggs, only illustrated as a numerical way of correcting differences. A modern explanation of the method is given.

Chapter fourteen
To Find the number agreeing with a given Logarithm.

The rest of the book is concerned with applications of Logarithms.

Chapter fifteen
To find the missing number of four numbers in proportion.

Chapter sixteen
To find a root of a given number.

Chapter seventeen
To find any number in a series of numbers in continued proportion; i.e. the intermediate terms in a G.P. ; Finance problems involving repayment of interest, etc.

Chapter eighteen
Uses of logs in solving for the various parameters of mainly right-angled triangles, given the sides.

Chapter nineteen
Eight problems concerning right-angled triangles are solved.

Chapter twenty
About a given base, to describe a triangle isoperimetric and of equal area to a given triangle.

Chapter twenty-one
A theorem of Apollonius is demonstrated geometrically and numerically by Briggs.

Chapter twenty-two
For a given base, the difference of the legs, and the area of the triangle, to find the legs of the triangle.

Chapter twenty-three
To find a triangle for which the area is equal to the perimeter.

Chapter twenty-four

Chapter twenty-five
Area and perimeter of Circle; surface area and volume of sphere.

Chapter twenty-six
Concerning ellipses, spheroids, and cask gauging.

Chapter twenty-seven
To divide a line according to the mean and extreme ratio. [i.e. the Fibonacci Numbers].

Chapter twenty-eight
To find the sides and areas of regular figures inscribed in a given circle. Including 3-, 4-, 5-, 6-, 8-, 10-, 12-, and 16-gon.

Chapter twenty-nine
Concerning the regular 7- and 9-, 15-, 24-, and 30-gons.

Chapter thirty
Concerning isoperimetric regular figures.

Chapter thirty-one
Concerning regular figures of the same area.

Chapter thirty-two
Concerning the five Platonic solids.

Ian Bruce February 2004