Ian Bruce

University of Adelaide, Australia

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.

PDF files require Adobe Acrobat Reader software - follow the link to download a free copy.

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

Constructing cyclic quadrilaterals.

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

The URL of this page is:

http://www-history.mcs.st-andrews.ac.uk/Miscellaneous/Briggs/index.html