“…nothing is more tedious, fellow mathematicians, in the practice of the mathematical arts, than the great delays suffered in the tedium of lengthy multiplications and divisions, the finding of ratios, and in the extraction of square and cube roots…The book contains fifty-seven pages of explanatory matter and ninety pages of tables of trigonometric functions and their natural logarithms. These tables greatly simplified calculations inith The Ith () is a ridge in Germany's Central Uplands which is up to 439 m high. It lies about 40 km southwest of Hanover and, at 22 kilometres, is the longest line of crags in North Germany. Geography Location The Ith is immediatel ...the many slippery errors that can arise…I have found an amazing way of shortening the proceedingsn which N, or n, is the fourteenth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''en'' (pronounced ), plural ''ens''. History ...all the numbers associated with the multiplications, and divisions of numbers, and with the long arduous tasks of extracting square and cube roots are themselves rejected from the work, and in their place other numbers are substituted, which perform the tasks of these rejected by means of addition, subtraction, and division by two or three only.”
The tables
In Napier’s time,The Descriptio text
Napier's ''Descriptio'' is divided into two books. The first describes his invention and some applications and includes the tables. The second discusses applications to trigonometry.Book I
Chapter 1 contains a series of definitions and propositions that explains Napier's conception of logarithms. He conceived logarithms in terms of two models of motion. In the first, a particle starts at a point and moves along a straight line at a constant speed. In the second, a particle moves along a straight line, starting with the same initial speed, but its speed decreases in proportion to its distance from the starting point. The logarithm of a number, ''a,'' is then the distance traveled by a particle in the constant speed model, during the time it takes the particle in the second, inverse proportional model to reach ''a.'' He defines the logarithm of 10,000,000 to be zero and that the logarithm of values less than that to be positive, while greater numbers have a negative logarithm, a reversal of sign from modern logarithms. (This sign change is sometimes expressed by saying Napier used logarithms with base 1/''e''.) He notes that he has freedom to choose any value to have a zero log, in modern terms the base, but chooses 10,000,000 for ease of calculation as it matches the "total sine" (the hypotenuse), in his sine tables. ''See'' Naperian logarithm. The second chapter describes the properties of logarithms and give some formulas (in text form) for working with ratios. It ends with a note stating he is delaying publication of his work on constructing logarithms until he sees how his invention is received. Chapter 3 describes the tables and their seven columns, see above. The fourth chapter explains how to use the tables and gives worked examples for sines tangents and secants. He also explains how to get logarithms of numbers directly by using the sine values as the argument and the log-sine values as result and vice versa. He discusses how to deal with different multiples of ten and introduces a notation, similar to modernBook II
Book II deals with "that noble kind of Geometry, that is called Trigonometry." The first chapter deals with using logarithms to solve problems in plane trigonometry with right triangles and, in particular, with small angles, where his trigonometric logarithms become large. The next chapter cover plane oblique triangles. The remaining chapters cover spherical trigonometry, starting with quadrants. He also describes hisMirifici Logarithmorum Canonis Constructio
Napier was reluctant to publish the theory and details of how he created his table of logarithms pending feedback from the mathematics community on his ideas, and he died shortly after publication of the ''Discriptio.'' His son Robert published the ''Constructio'' in 1619. The volume has a preface by Robert and several appendices, including a section on John Napier’s methods for more easily solving spherical triangles, and a section by Henry Biggs on “another and better kind of logarithms,” namely base 10 or common logarithms. An English translation by William Rae Macdonald was published, with annotations, in 1889. Napier‘s describes logarithms via a correspondence between two points moving under different speed profiles. The first point, P, moves along a finite line segment P0 to Q, with an initial speed that decreases proportional to P's distance to Q. The second point, L, moves along an unbounded line segment starting at L0 at the same time as P and with the same initial speed, but maintaining that speed without change. For each possible position of P, measured by its distance from Q, there is a corresponding simultaneous position of L. Napier defined the logarithm of the distance from P to Q to be distance from L0 to that L.Computation approach
Napier relies on several insights to compute his table of logarithms. To achieve high accuracy he starts with a large base of 10,000,000. But he then gets additional precision by using decimal fractions in a notation that he invented, but now universally familiar, namely using aCalculating first logarithm
Using his two line model, Napier finds lower and upper bounds for the logarithm of .9999999. His lower bound assumes the point P does not slow down, in which case L will move a distance of 1-.9999999. His upper bound assumes P started out at its final speed of 0.9999999 in which case L will have moved the distance of 1/0.9999999. Scaled up by his radius of 10,000,000, the lower bound is 1 and his upper bound is 1.0000001. He suggests that since the difference between these values is tiny, any value between them will present an "insensible error" of less than one part in 10 million, but he chooses, without much explanation, the midpoint, 1.00000005. This choice gives him far greater precision, as his translator, William Rae Macdonald. points out in an appendix, noting that Napier's scaled up value for the logarithm of .9999999 is very close to the correct value, 1.000000050000003333333583..., and that all his subsequent computations of logarithms derive from the 1.00000005 value. Macdonald suggests that Napier must have had a better reason for picking the midpoint.Auxiliary tables
Napier uses these insights to construct three tables. The first table, in modern notation, consists of the numbers 10000000*(0.9999999)''n'' for ''n'' ranging from 0 to 100. The second consists of the numbers 100000*(0.99999)''n'' for ''n'' ranging from 0 to 50. He then applies his value for the log of .9999999 to fill in logarithms for all the entries in his first table. He can use the last entry to compute the log of .99999, since .9999999100 is very close to .99999. He then the uses his second table, which is essentially 50 powers of .99999 to compute the logarithm of .9995. Macdonald also points out that an error crept into Napier's calculation of the second table; Napier's fiftieth value is 9995001.222927, but should be 9995001.22480. Macdonald discusses the consequences of this error in his appendix.The radical table
Napier then constructs a third table of proportions with 69 columns and 21 rows, which he calls his "radical table." The proportion along the top rows, starting with 1 is 0.99. The entries in each column are in proportion 0.9995. (Note that 0.9995 = 1-1/2000, allowing "tolerably easy" multiplication by halving, shifting and subtracting.) Napier uses the first column to computing the logarithm of .99, using log of .9995, which he already has. He can now fill in the logarithm of each entry in the third table because, by proportionality, the difference in logarithms between entries is constant. The third table now provides logarithms for a set of 1,449 values that cover the range from roughly 5,000,000 to 10,000,000, which corresponds to values of the sine function from 30 to 90 degrees, assuming a radius of 10,000,000. Napier then explains how to use the tables to calculate a bounding interval for logarithms in that range.Constructing the published tables
Napier then gives instructions for reproducing his published tables, with their seven columns and coverage of each minute of arc. He does not compute the sines themselves, the values for which are to be filled in from an already available table. " Reinhold's common table of sines, or any other more exact, will supply you with these values." Logarithms of sines for angles from 30 degrees to 90 degrees are then computed by finding the closest number in the radical table and its logarithm and calculating the logarithm of the desired sine by linear interpolation. He suggests several ways for computing logarithms for sines of angles less than 30 degrees. For example, one can multiply a sine that is less than 0.5 by some power of two or ten to bring it into the range .5,1 After finding that logarithm in the radical table, one adds the logarithm of the power of two or ten that was used (he gives a short table), to get the required logarithm. Napier ends by pointing out that two of his methods for extending his table produce results with small differences. He proposes that others “who perchance may have plenty of pupils and computers” construct a new table with a larger scale factor of 10,000,000,000, by the same methods but using a radical table with only 35 columns, enough to cover angels from 45 to 90 degrees.After matter
In an appendix, Napier discusses construction of “another and better kind of logarithm” where the logarithm of one is zero and the logarithm of ten is 10,000,000,000, the index. This is essentially base 10 logarithms with the large scale factor. He discusses various ways to compute such a table and ends by describing the logarithm of 2 as the number of digits in 210,000,000, which he computes as 301029995. The appendix is followed by remarks Henry Briggs on Napier's concepts and base-10 logarithms. The next section is a 12 page essay by Napier titled “Some very remarkable propositions for the solution of spherical triangles with wonderful ease,” where he describes how to solve them without dividing them into two right triangles. This section is also followed by commentary from Briggs. The translator, Macdonald, includes some notes at this point, discussing the spelling of Napier’s name, references to delays in publishing the second volume, the development of decimal arithmetic, the error in Napiers second table and the accuracy of Napier’s method, and methods for computing base-10 logarithms. The last section is a catalog by Macdonald of Napier's works in public libraries, including religious works, editions in different languages. and other books related to the work of John Napier and logarithms.Reception
Napier's novel method of calculation spread quickly in Britain and abroad. Kepler dedicated his 1620 ''Ephereris'' to Napier, with a letter congratulating him on his invention and its benefits to astronomy. Keppler found no essential errors except for some inaccuracies at small angles. Edward Wright, an authority on celestial navigation, translated Napier's Latin ''Descriptio'' into English in 1615, the next year, though publication was delayed by Wright's death. Briggs extended the concept to the more convenient base 10, or common logarithm. Ursinus called Napier "a mathematician without equal." Three hundred years later, in 1914, E. W. Hobson called logarithms "one of the very greatest scientific discoveries that the world has seen." In 1620See also
*References
{{reflist Logarithms History of mathematics