History of Computers and Computing, Calculating tools, Logarithms natural numbers 1 2 4 8 16 32 64 128 256 512 1024 . And the series below is an arithmetic one whose values are the corresponding base 2 logarithms: logarithms 0 1 2 3 4 5 6 7 8 9 10 . It had long been known that if you take any two numbers in the arithmetic progression, say 3 and 4, their sum, 7, would indicate the position of the term in the geometric series that is the product of the terms in the corresponding positions of the geometric series, e.g., 3 + 4 = 7 and 8 x 16 = 128 (the third times the fourth = the seventh). This is starting to look very much like our own conception of logarithms as being the powers to which some base number is raised, a concept that was not understood in Napier's time. Often the use of a good form of notation will suggest some basic mathematical principle.

Our use of indices to indicate the power to which a number is being raised seems to have an obvious connection with logarithms, but without this form of notation, the connection is vague at best. John Napier came at the idea of logarithms not by algebra and indices but by way of geometry. When first thinking about this subject, he used the term artificial number but later created the term logarithm from two Greek words—λoγoς and αριθμός, meaning word, ratio and number respectively. He decided to use this term because his logarithms were based on the concept of points moving down lines in which the velocity of one point was based on the ratio of the lengths of the line on either side of it. In the end of 1614 one of the most famous English mathematicians of the day, Henry Briggs (1561-1631), who was a Professor of Geometry at Gresham College, London, obtained a copy of Napier's Descriptio and, by March of the following year wrote that: Napier, lord of Markinston, hath set my head and hands at work with his new and admirable logarithms.

I hope to see him this summer, if it please God; for I never saw a book which pleased me better, and made me more wonder . Briggs immediately began to popularize the concept of logarithms in his lectures and even began to work on a modified version of the tables. Working together with Napier for some time (he was able to visit Napier twice at his baronial estate), later on Briggs will propose that the base of the logarithms should be changed in order to make them easier to use, thus we have 10 based logarithms, and in 1624 published tables, containing the logs of the numbers from 1 to 20,000 and from 90,000 to 100,000 all calculated to 14 decimal places. In order to make calculations by means of logarithms, we have to use tables with logarithms. If we have to multiply two numbers, we have to find their logarithms in the table, to add the logarithms, and then to find in the table the number, which logarithm corresponds to the sum of logarithms. But what to do, if this logarithm cannot be find in the table?

Let's for example suppose, that we have only a table with decimal logarithms of the integers up to 1000, but we need a logarithm of a fractional number, e.g. 7,93. As long as log10(7,93)=log10(793)–log10(100) So, needed logarithm can be found by subtraction of 2 (this is the decimal logarithm of 100) from the logarithm of 793. Of course, the usefulness of the decimal logarithms is due to our decimal numbering system. It was not only Briggs who was impressed by Napier's Descriptio. In 1617, Johann Kepler first saw the Descriptio in Prague.

He was too busy to pay it much attention, as during this period that he was hard at work on his third law of planetary motion, but did acknowledge its existence in a letter to his friend Wilhelm Schickard , the creator of the first mechanical calculator in the world , where he indicated: A Scottish baron has started up, his name I cannot remember, but he has put forth some wonderful mode by which all necessity of multiplications and divisions are commuted to mere additions and subtractions. A year later, Kepler wrote to Napier expressing his admiration and letting him know that he must publish the promised Constructio as soon as possible. Unfortunately, Napier had been dead for two years before the letter arrived. The letter may have spurred John's son, Robert Napier, into putting the finishing touches on his father's notes and overseeing the publication of the Constructio in 1620. Within twenty years of the time that Briggs's tables first appeared, the use of logarithms had spread worldwide. From being a limited tool of great scientists like Johann Kepler, they had become commonplace in the schoolrooms of the civilized nations. Logarithms were used extensively in all trades and professions that required calculations to be done.

It is hard to imagine an invention that has helped the process of computation more dramatically than has logarithms. Very soon after popularizing of logarithms began attempts for removing of the nasty end error-prone process of looking into tables and manual adding of numbers. First was a colleague of Brigs and professor of astronomy at Gresham College—Edmund Gunter (1581-1626). Briggs's work naturally came to the notice of Gunter, who had some earlier experience in the development of calculating instruments, having been one of the major figures in the perfection of an instrument known as a sector. This experience soon led him to realize that the process of adding together a pair of logarithms could be partially automated by engraving a scale of logarithms on a piece of wood and then using a pair of compasses to add together two values in much the same way, as he would have done when using a sector. Not only did this method eliminate the mental work of addition, but it also removed the necessity for the error-prone and time-consuming process of looking up the logarithms in a table.

Gunter described his instrument in the book Description and use of the Sector, the Crosse-staffe and other Instruments, published in 1623. Such a rule is frequently referred to as a Gunter line or Gunter Scale, often combination not only of a logarithmic scale, but also with chords, sines, tangents and rhumbs. A two foot long boxwood ruler inscribed with a variety of scales was a standard navigator's tool up until the end of the 19th century. The original Drawing of Gunter Scale from 1624 (upper) and part of a wooden Gunter's Scale (lower image) After Gunter, several English mathematicians almost together invented some kind of rules. First apparently was a friend of Brigs and Gunter—William Oughtred (1574-1660), a priest and one of the leading mathematicians of his day.

Oughtred noted that Gunter's Line of Numbers required a pair of dividers in order to measure off the lengths of logarithmic values along the scale and quickly came up with the idea that, if he had two such scales marked along the edges of the pieces of wood, he could slide them relative to each other and thus do away with the need for a pair of dividers. He also saw that if there were two disks, one slightly smaller than the other, with a Line of Numbers engraved around the edge of each, that they could be pinioned together at their centers and rotated relative to one another to give (so called Circles of Proportion) the same effect as having Gunter's scale engraved on two bits of wood (see the lower drawing). Oughtred invented the rectangular movable slide rule and the circular slide rule around 1622, but published the description as late as 1632 (in the book The Circles of Proportion and the Horizontal instrument), thus giving the chance of some other others to outstrip him.

Oughtred's Circles of Proportion Oughtred probably described his devices to one of his pupils (Oughtred taught mathematics at his residence, without compensation, to promising pupils), Richard Delamain (1600-1644), who was a teacher of mathematics, living and working in London (he was a private tutor of mathematics to Charles I, King of Great Britain and Ireland). Delamain, described a circular slide rule in a 32 page pamphlet Grammelogia which was sent to the King in 1629 and published the following year. This led to verbal disputes between Oughtred and Delamain, as each combatant accuses the other of stealing the invention of the circular slide rule, The first description of the Gunter's logarithmic scale in the form of a plain scale was published in 1624 in Paris by the mathematician Edmund Wingate (1596-1656). Wingate included a copperplate engraving, measuring 25 inches in length, of Gunter's scales. In 1628 Wingate published an English translation of this work with the title The Construction and Use of the Line of Proportion....

This book contains a double scale, on one side of which is a logarithmic scale, on the other a tabular scale. One simply reads the logarithm of a number on the logarithmic scale by examining the proportional scale next to it. This is not yet a contemporary slide rule, since it has no moving parts and is not a Gunter line, since one does not perform calculations with the aid of pair of compasses: it is simply a diagrammatic substitute for a table of logarithms. The earliest known dated slide rule (which uses wooden scales bound together with metal bands), is the rule of Robert Bissaker, manufactured in 1654 (see the nearby photo). It is about 60 cm in length, with section about 6 sq. cm and consists of 3 wooden scaled plates, as the 2 outer plates are fixed by means of a copper frame, while the inner plate can be slide between them (see the lower photo). It seems that the idea of the slider (called also cursor or runner), important element of a contemporary slide rule, first came to the great scientist Isaac Newton (1643-1727).

In 1675 Newton demonstrated a way for solving of equations from the third degree, using 1 unmovable and 3 movable logarithmic scales and a slider. As a element of the slide rule however, the slider appeared 100 years late in the rule of mathematician John Robertson (1707–1776). In 1775 he added the cursor or runner, which allowed setting to be transferred to any of several parallel scales, as well as holding a position while the slide was moved. The English inventor James Watt, known for his work on the steam engine, was responsible, at least in part, for one of the first really well-made slide rules in the end of 18th century. He devised a simple slide rule consisting of one sliding piece between two fixed stocks (a design that had been in use for a considerable period of time), carefully engraved the face with four basic scales, and put tables of various constants on the back. His rule was accurate enough that others soon requested copies for themselves and Watt manufactured this so-called Soho Slide Rule for several years (see the lower photo).

James Watt's slide rule Soho Victor Mayer Amédée Mannheim (1831-1906), an officer of the French artillery, invented in 1850s what may be considered the first of the modern slide rules. He designed a very simple slide rule much like that manufactured by Watt, but added the movable double-sided cursor, which is a integral part of the slide rule today. Mannheim's design was adopted as the standard for the French artillery and, after a few years, examples of it began to appear in other countries. Once established, the progress of the slide rule was extremely rapid. Many different forms were produced by several different major manufacturers. The number of scales to be found on each instrument increased to the point that eighteen or twenty different scales were regularly engraved on the better quality instruments. Both sides of the rule were used and the center, sliding portion could often be turned over or completely replaced to provide even more combinations of scales. Special slide rules incorporating such things as a scale of atomic and molecular weights were created for chemists, and almost any branch of science or engineering could boast that at least one manufacturer produced a slide rule designed for their particular use.

The accuracy of the slide rule was improved by several people who modified the basic form so that the logarithmic scales were wrapped around cylinders or into spirals. One device, known as Fuller's Slide Rule (see the lower figure), invented by the Professor of Engineering at Queen’s College Belfast George Fuller in 1860s (US patent 219246 from 1879), was equivalent to a standard slide rule over eighty-four feet long, yet could be easily held in the hand. It was possible to work correctly to four figures, and sometimes even five, with this particular unit. Fuller's Slide Rule A similar as Fuller's slide rule was made by Edwin Thacher of Pittsburg, Pennsylvania in 1870s. Placing the logarithmic scale on a drum and series of crosspieces, the Thacher functioned like an 18-meter slide rule for calculations up to five significant digits. Inventor Thacher patented it in 1881 (see US patent 249117 ). This one (see the lower photo) was manufactured by Keuffel & Esser of New York around 1887.

Thacher's Slide Rule (1887) Mathematical Table - Encyclopedia Mathematical Table GEOGRAPHICAL NAMES MATHEMATICAL. TABLE In any table the results tabulated are termed the " tabular results " or " respondents, " and the corresponding numbers by which the table is entered are termed the " arguments." A table is said to be of single or double entry according as there are one or two arguments. For example, a table of logarithms is a table of single entry, the numbers being the arguments and the logarithms the tabular results; an ordinary multiplication table is a table of double entry, giving xy as tabular result for x and y as arguments. The intrinsic value of a table may be estimated by the actual amount of time saved by consulting it; for example, a table of square roots to ten decimals is more valuable than a table of squares, as the extraction of the root would occupy more time than the multiplication of the number by itself. The value of a table does not depend upon the difficulty of calculating it; for, once made, it is made for ever, and as far as the user is concerned the amount of labour devoted to its original construction is immaterial.

In some tables the labour required in the construction is the same as if all the tabular results had been calculated separately; but in the majority of instances a table can be formed by expeditious methods which are inapplicable to the calculation of an individual result. This is the case with tables of a continuous quantity, which may frequently be constructed by differences. The most striking instance perhaps is afforded by a factor table or a table of primes; for, if it is required to determine whether a given number is prime or not, the only universally available method (in the absence of tables) is to divide it by every prime less than its square root or until one is found that divides it without remainder. But to form a table of prime numbers the process is theoretically simple and rapid, for we have only to range all the numbers in a line and strike out every second number beginning from 2, every third beginning from 3, and so on, those that remain being primes. Even when the tabular results are constructed separately, the method of differences or other methods connecting together different tabular results may afford valuable verifications. By having recourse to tables not only does the computer save time and labour, but he also obtains the certainty of accuracy.

The invention of logarithms in 1614, followed immediately by the calculation of logarithmic tables, revolutionized all the methods of calculation; and the original work performed by Henry Briggs and Adrian Vlacq in calculating logarithms in the early part of the 17th century has in effect formed a portion of every arithmetical operation that has since been carried out by means of logarithms. And not only has an incredible amount of labour been saved, 1 but a vast number of calculations and researches have been rendered practicable which otherwise would have been beyond human reach. The mathematical process that underlies the tabular method of obtaining a result may be indirect and complicated; for example, the logarithmic method would be quite unsuitable for the multiplication of two numbers if the logarithms had to be calculated specially for the purpose and were not already tabulated for use.

The arrangement of a table on the page and all typographical details-such as the shape of the figures, their spacing, the thickness and placing of the rules, the colour and quality of the paper, &c.- are of the highest importance, as the computer has to spend hours with his eyes fixed upon the book; and the efforts of eye and brain required in finding the right numbers amidst a mass of figures on a page and in taking them out accurately, when the computer is tired as well as when he is fresh, are far more trying than the mechanical action of simple reading. Moreover, the trouble required by the computer to learn the use of a table need scarcely be considered; the important matter is the time and labour saved by it after he has learned its use. In the following descriptions of tables an attempt is made to give an account 'of all those that a computer of the present day is likely to use in carrying out arithmetical calculations. Tables relating to ordinary arithmetical operations are first described, and afterwards an account is given of the most useful and least technical of the more strictly mathematical tables, such as factorials, gamma functions, integrals, Bessel's functions, &c.

Nearly all modern tables are stereotyped, and in giving their titles the accompanying date is either that of the original stereotyping or of the tirage in question. In tables that have passed through many editions the date given is that of the edition described. A much fuller account of general tables published previously to 1872, by the present writer, is contained in the British Association Report for 1873, pp. 1-175. Tables of Divisors (Factor Tables) and Tables of Primes.-The existing factor tables extend to 10,000,000. In 1811 L. Chernac published at Deventer his Cribrum arithmeticum, which gives all the prime divisors of every number not divisible by 2, 3, or 5 up to 1,020,000. In1814-1817J. C. Burckhardt published at Paris his Tables des diviseurs, giving the least divisor of every number not divisible by 2, 3, or 5 up to 3,036,000.

The second million was issued in 1814, the third in 1816, and the first in 1817. The corresponding tables for the seventh, eighth, and ninth millions were calculated by Z. Dase and issued at Hamburg in 1862, 1863, and 1865. Dase died suddenly in 1861 during the progress of the work, and it was completed by H. Rosenberg. Dase's calculation was performed at the instigation of Gauss, and he began at 6,000,000 because the Berlin Academy was in possession of a manuscript presented by Crelle extending Burckhardt's tables from 3,000,000 to 6,000,000. This manuscript was found on examination to be so inaccurate that the publication was not desirable, and accordingly the three intervening millions were calculated and published by James Glaisher, the Factor Table for the Fourth 1 Referring to factor tables, J. H. Lambert wrote (Supplementa tabularum, 1798, p.

xv.): " Universalis finis talium tabularum est ut semel pro semper computetur quod saepius de novo computandum foret, et ut pro omni casu computetur quod in futurum pro quovis casu computatum desiderabitur." This applies to all tables. Million appearing at London in 1879, and those for the fifth and sixth millions in 1880 and 1883 respectively (all three millions stereotyped). The tenth million, though calculated by Dase and Rosenberg, has not been published. The nine quarto volumes (Tables des diviseurs, Paris, 1814-1817; Factor Tables, London, 1879-1883; Factoren-Tafeln, Hamburg, 1862-1865) thus form one uniform table, giving the least divisor of every number not divisible by 2, 3, or 5, from unity to nine millions.

The arrangement of the results on the page, which is due to Burckhardt, is admirable for its clearness and condensation, the least factors for 9000 numbers being given on each page. The tabular portion of each million occupies 112 pages. The first three millions were issued separately, and also bound in one volume, but the other six millions are all separate. Burckhardt began the publication of his tables with the second million instead of the first, as Chernac's factor table for the first million was already in existence. Burckhardt's first million does not supersede Chernac's, as the latter gives all the prime divisors of numbers not divisible by 2, 3, or 5 up to 1,020,000. It occupies 1020 pages, and Burckhardt found it very accurate; he detected only thirty-eight errors, of which nine were due to the author, the remaining twenty-nine having been caused by the slipping of type in the printing. The errata thus discovered are given in Burckhardt's first million. Other errata are contained in Allan Cunningham's paper referred to below.

Burckhardt gives but a very brief account of the method by which he constructed his table; and the introduction to Dase's millions merely consists of Gauss's letter suggesting their construction. The Introduction to the Fourth Million (pp. 52) contains a full account of the method of construction and a history of factor tables, with a bibliography of writings on the subject. The Introduction (pp. 103) to the Sixth Million contains an enumeration of primes and a great number of tables relating to the distribution of primes in the whole nine millions, portions of which had been published in the Cambridge Philosophical Proceedings and elsewhere. A complete list of errors in the nine millions was published by J. P. Gram (Acta mathematics, 18 93, 1 7, p. 310). These errors, 141 in number, and which affect principally the second, third, eighth, and ninth millions, should be carefully corrected in all the tables. In 1909 the Carnegie Institution of Washington published a factor table by Prof. D. N. Lehmer which gives the least factor of all numbers not divisible by 2, 3, 5, or 7, up to ten millions.

This table, which covers a range of 21,000 numbers on a single page, was reproduced by photography from a typewritten copy of the author's original manuscript. The introduction contains a list of errata in the nine millions previously published, completely confirming Gram's list. The factor tables which have just been described greatly exceed both in extent and accuracy any others of the same kind, the largest of which only reaches 408,000. This is the limit of Anton Felkel's Tafel aller einfachen Factoren (Vienna, 1776), a remarkable and extremely rare book,' nearly all the copies having been destroyed. Georg Vega (Tabulae, 1797) gave a table showing all the divisors of numbers not divisible by 2, 3, or 5 up to 502,000, followed by a list of primes from 102,000 to 400,313. In the earlier editions of this work there are several errors in the list, but these are no doubt corrected in J. A. Hiilsse's edition (1840). J.

Salomon (Vienna, 1827) gives the least divisor of all numbers not divisible by 2, 3, or 5, up to 102,011, and B. Goldberg (Primzahlen and Factoren-Tafeln, Leipzig, 1862) gives all factors of numbers not divisible by 2, 3, or 5 up to 251,650. H. G. Kohler (Logarithmisch-trigonometrisches Handbuch, 1848 and subsequent editions) gives all factors of numbers not prime or divisible by 2, 3, 5, or II up to 21,525. Peter Barlow (Tables, 1814) and F. Schaller (Primzahlen-Tafel, Weimar, 1855) give all factors of all numbers up to 10,000. Barlow's work also contains a list of primes up to 100,103. Both the factor table and the list of primes are omitted in the stereotyped (1840) reprint.

Full lists of errata in Chernac (181I), Barlow (1814), Hiilsse's Vega (1840), Kohler (1848), Schaller (1855), and Goldberg (1862) are contained in a paper by Allan Cunningham (Mess. of Math., 1904, 34, P. 24; 1905, 35, p. 24). V. A. Le Besgue (Tables diverses pour la decomposition des nombres, Paris, 1864) gives in a table of twenty pages, the least factor of numbers not divisible by 2, 3, or 5 up to 115,500. In Rees's Cyclopaedia (1819), article " Prime Numbers," there is a list of primes to 217,219 arranged in decades. The Fourth Million (1879) contains a list of primes up to 30,341. The fourth edition of the Logarithmic Tables (London, and Ithaca, N.Y., 1893) of G. W.

Jones of Cornell University contains a table of all the factors of numbers not divisible by 2 or 5 up to 20,000. In the case of primes the ten-place logarithm is given. This table does not occur in the third edition (Ithaca, N.Y., 1891). On the first page of the Second Million Burckhardt gives the first nine multiples of the primes to 1423; and a smaller table of the same kind, extending only to 313, occurs in Lambert's Supplementa (1798). Several papers contain lists of high primes (i.e. beyond the range of the 2 For information about it, see a paper on " Factor Tables," in Camb. Phil. Proc. (1878), iii. 99-238, or the Introduction to the Fourth Million. factor tables). Among these may be mentioned two, by Allan Cunningham and H. J. Woodall jointly, in the Mess. of Math., 1902, 31, p. 165; 1905, 34, p. 72.

See also the papers on factorizations of high numbers referred to under Tables relating to the Theory of Numbers. The Vienna Academy possesses the manuscript of an immense factor table extending to 100,000,000, constructed many years ago by J. P. Kulik (1793-1863) (see Ency. math. Wiss., 1900-1904, i. 952, and Lehmer's Factor Table, p. ix.). Multiplication Tables.-A multiplication table is usually of double entry, the two arguments being the two factors; when so arranged it is frequently called a Pythagorean table. The largest and most useful work is A. L. Crelle's Rechentafeln (Bremiker's edition, 1857, stereotyped; many subsequent editions with German, French, and English title-pages), which gives in one volume all the products up to 1000 X 1000, so arranged that all the multiples of any one number appear on the same page. The original edition was published in 1820 and consisted of two thick octavo volumes.

The second (stereotyped) edition is a convenient folio volume of 450 pages.' In 1908 an entirely new edition, edited by O. Seeliger, was published in which the multiples of 10, 20, ..., 990 (omitted in previous editions) are included. This adds 50 pages to the volume, but removes what has been a great drawback to the use of the tables. Other improvements are that the tables are divided off horizontally and vertically by lines and spaces, and that, for calculations in which the last two figures are rejected, a mark has been placed to show when the last figure retained should be increased. Two other tables of the same extent (moo X moo), but more condensed in arrangement, are H. C. Schmidt's Zahlenbuch (Aschersleben, 1896), and A. Henselin's Rechentafel (Berlin, 1897). An anonymous table, published at Oldenburg in 1860, gives products up to 500X509, and M.

Cordier, Le Multiplicateur de trois cents carres (Paris, 1872), gives a multiplication table to 300X300 (intended for commercial use). In both these works the product is printed in full. The four following tables are for the multiplication of a number by a single digit. (I) A. L. Crelle, Erleichterungstafel fur jeden, der zu rechnen hat (Berlin, 1836), a work extending to woo pages, gives the product of a number of seven figures by a single digit, by means of a double operation of entry. Each page is divided into two tables: for example, to multiply 93 $2 477 by 7 we turn to page 825, and enter the right-hand table at line 77, column 7, where we find 77339; we then enter the left-hand table on the same page at line 93, column 7, and find 656, so that the product required is 6567-7339. (2) C. A.

Bretschneider, Produktentafel (Hamburg and Gotha, 1841), is somewhat similar to Crelle's table, but smaller, the number of figures in the multiplicand being five instead of seven. (3) In S. L. Laundy, A Table of Products (London, 1865), the product of any five-figure number by a single digit is given by a double arrangement. The extent of the table is the same as that of Bretschneider's, as also is the principle, but the arrangement is different, Laundy's table occupying only to pages and Bretschneider's 99 pages. (4) G. Diakow's Multiplikations-Tabelle (St Petersburg, 1897) is of the same extent as Bretschneider's table but occupies woo pages. Among tables extending to Ioo X moo (i.e. giving the products of two figures by three) may be mentioned C. A. Muller's Multiplications-Tabellen (Karlsruhe, 1891). The tables of L. Zimmermann (Rechentafeln, Liebenwerda, 1896) and J.

Riem (Rechentabellen fur Multiplication, Basel, 1897) extend to 100X10,000. In a folio volume of 500 pages J. Peters (Rechentafeln fur Multiplikation and Division mit einbis vierstelligen Zahlen, Berlin, 1909) gives products of four figures by two. The entry is by the last three figures of the multiplicand, and there are 2000 products on each page. Among earlier tables, the interest of which is mainly historical, mention may be made of C. Hutton's Table of Products and Powers of Numbers (London, 1781), which containsa table up to too X 1000, and J. P. Gruson's Grosses Einmaleins von Eins bis Hunderttausend (Berlin, 1799)-a table of products up to 9 X to,000. The author's intention was to extend it to 100,000, but only the first part was published.

In this book there is no condensation or double arrangement; the pages are very large, each containing 125 lines. Quarter-Squares.-Multiplication may be performed by means of a table of single entry in the manner indicated by the formula ab =±(a+b)2-±(a-b)'. ' Only one other multiplication table of the same extent as Crelle's had appeared previously, viz. Herwart von Hohenburg's Tabulae arithmeticae lrporeackupkQews universales (Munich, 1610), a huge folio volume of more than a thousand pages. It appears from a correspondence between Kepler and von Hohenburg, which took place at the end of 1608, that the latter used his table when in manuscript for the performance of multiplications in general, and that the occurrence of the word prosthaphaeresis on the title is due to Kepler, who pointed out that by means of the table spherical triangles could be solved more easily than by Wittich's prosthaphaeresis. The invention of logarithms four years later afforded another means of performing multiplications, and von Hohenburg's work never became generally known.

On the method of prosthaphaeresis, see Napier, John, and on von Hohenburg's table, see a paper " On multiplication by a Table of Single Entry," Phil. Mag., 1878, ser. v., 6, p. 331. Thus with a table of quarter-squares we can multiply together any two numbers by subtracting the quarter-square of their difference from the quarter-square of their sum. The largest table of quarter-squares is J. Blater's Table of Quarter-Squares of all whole numbers from z to 200,000 (London, 1888), 2 which gives quarter-squares of every number up to 200,000 and thus yields directly the product of any two five-figure numbers. This fine table is well printed and arranged. Previous to its publication the largest table was S. L. Laundy's Table of Quarter-Squares of all numbers up to I oo,000 (London, 1856), which is of only half the extent, and therefore is only directly available when the sum of the two numbers to be multiplied does not exceed too,000.

Smaller works are J. J. Centnerschwer, Neuerfundene Multiplicationsand Quadrat-Tafeln (Berlin, 1825), which extends to 20,000, and J. M. Merpaut, Tables arithmonomiques (Vannes, 1832), which extends to 40,000. In Merpaut's work the quarter-square is termed the " arithmone." L. J. Ludolf, who published in 1690 a table of squares to 100,000 (see next paragraph), explains in his introduction how his table may be used to effect multiplications by means of the above formula; but the earliest book on quartersquares is A. Voisin, Tables des multiplications, ou logarithmes des nombres entiers depuis I jusqu'a 20,000 (Paris, 1817). By a logarithm Voisin means a quarter-square, i.e. he calls a a root and 4a' its logarithm. On the subject of quarter-squares, &c., see Phil. Mag. [v.] 6, p. 331.

Squares, Cubes, & g c., and Square Roots and Cube Roots.-The most convenient table for general use is P. Barlow's Tables (Useful Knowledge Society, I r ondon, from the stereotyped plates of 1840), which gives squares, cubes, square roots, cube roots, and reciprocals to t 0,000. These tables also occur in the original edition of 1814. The largest table of squares and cubes is J. P. Kulik, Tafeln der Quadratand Kubik-Zahlen (Leipzig, 1848), which gives both as far as 100,000. Blater's table of quarter-squares already mentioned gives squares of numbers up to 100,000 by dividing the number by 2; and up to 200,000 by multiplying the tabular result by 4. Two early tables give squares as far as 100,000, viz.

Maginus, Tabula tetragonica (Venice, 1592), and Ludolf, Tetragonometria tabularia (Amsterdam, 1690); G. A. Jahn, Tafel der Quadratund Kubikwurzeln (Leipzig, 1839), gives squares to 27,000, cubes to 24,000, and square and cube roots to 25,500, at first to fourteen decimals and above 1010 to five. E. Gelin (Recueil de tables numeriques, Huy, 1894) gives square roots (to 15 places) and cube roots (to to places) of numbers up to boo. C. Hutton, Tables of Products and Powers of Numbers (London, 1781), gives squares up to 25,400, cubes to 10,000, and the first ten powers of the first hundred numbers. P. Barlow, Mathematical Tables (original edition, 1814), gives the first ten powers of the first hundred numbers.

The first nine or ten powers are given in Vega, Tabulae (1797), and in Hulsse's edition of the same (1840), in Kohler, Handbuch (1848), and in other collections. C. F. Faa de Bruno, Calcul des erreurs (Paris, 1869), and J. H. T. Muller, Vierstellige Logarithmen (1844), give squares for use in connexion with the method of least squares. Four-place tables of squares are frequently given in fiveand four-figure collections of tables. Small tables often occur in books intended for engineers and practical men. S. M. Drach (Messenger of Math., 1878, 7, p. 87) has given to 33 places the cube roots (and the cube roots of the squares) of primes up to 127. Small tables of powers of 2, 3, 5, 7 occur in various collections.

In Vega's Tabulae (1797, and the subsequent editions, including Hulsse's) the powers of 2, 3, 5 as far as the 45th, 36th, and 27th respectively are given; they also occur in Kohler's Handbuch (1848). The first 25 powers of 2, 3, 5, 7 are given in Salomon, Logarithmische Tafeln (1827). W. Shanks, Rectification of the Circle (1853), gives every 12th power of 2 up to 2 721. A very valuable paper (" Power-tables, Errata ") published by Allan Cunningham in the Messenger of Math., 1906, 35, p. 13, contains the results of a careful examination of 27 tables containing powers higher than the cube, with lists of errata found in each. Before using any power table this list should be consulted, not only in order to correct the errata, but for the sake of references and general information in regard to such tables. In an appendix (p.

23) Cunningham gives errata in the tables of squares and cubes of Barlow (1814), Jahn (1839), and Kulik (1848). Triangular Numbers.-E. de Joncourt, De natura et praeclaro usu simplicissimae speciei numerorum trigonalium (The Hague, 1762), contains a table of triangular numbers up to 20,000: viz. 2n(n+I) is given for all numbers from n=1 to 20,000. The table occupies 224 pages. Reciprocals.-P. Barlow's Tables (1814 and 1840) give reciprocals up to 10,000 to 9 or 10 places; and a table of ten times this extent is given by W. H. Oakes, Table of the Reciprocals of Numbers from I to 100,000 (London, 1865). This table gives seven figures of the reciprocal, and is arranged like a table of seven-figure logarithms, differences being added at the side of the page.

The reciprocal 2 The actual place of publication (with a German title, &c.) is Vienna. The copies with an English title, &c., were issued by Triibner; and those with a French title, &c., by Gauthier-Villars. All bear the date 1888. of a number of five figures is therefore taken out at once, and two more figures may be interpolated for as in logarithms. R. Picarte, La Division reduite a une addition (Paris, 1861), gives to ten significant figures the reciprocals of the numbers from io,000 to ioo,000, and also the first nine multiples of these reciprocals. J. C. Houzeau gives the reciprocals of numbers up to loo to 20 places and their first nine multiples to 12 places in the Bulletin of the Brussels Academy, 1875, 40, p. 107. E. Gain (Recueil de tables numeriques, Huy, 1894) gives reciprocals of numbers to loon to to places.

Table of contents Logarithms: The Early History of a Familiar Function Logarithms: The Early History of a Familiar Function Author(s): Kathleen M. Clark (The Florida State University) and Clemency Montelle (University of Canterbury) It may come as a surprise to many that often times mathematical concepts don't end up like they started! For those of you who think mathematics is timeless, fixed, and full of unchanging truths, such a proposition may seem unbelievable. But there are many instances in the history of mathematics of the development of a mathematical concept way beyond the purposes and potentialities that its original inventors intended. An example that will be familiar to you all is the logarithm. What is a logarithm? Ask a modern mathematician nowadays and you will get a very different answer from the one you might have got from a mathematician several centuries ago. Indeed, even the very first mathematicians who worked with the logarithmic relation would have given an explanation that would seem quite foreign to a modern mathematician. So how did the logarithmic relation come about, and how is it that the concept underwent so much change? We will address these questions by looking at the emergence of this concept, and examining some of the issues surrounding its origins.

In fact, the question of the origins of the logarithmic relation does not have a simple answer. At least two scholars, the Scottish baron John Napier (1550-1617) and Swiss craftsman Joost Bürgi (1552-1632), produced independently systems that embodied the logarithmic relation and, within years of one another, produced tables for its use. This parallel insight is fascinating and rich in historical detail, and it reveals some methodological challenges for historians of mathematics. In light of all this, we will examine the ideas of these two scholars, as well as explore how historians have portrayed this intricate situation and the questions it raises about mathematics. In large part, we intend to re-introduce teachers to a concept that is often taught without any reference to its original appearance on the mathematical scene. We hope that a close examination of Napier's and Bürgi's conceptions will enable teachers to consider alternative placement for introducing the idea of logarithms – as part of or after a unit on sequences.

Furthermore, we provide in what follows mathematical and historical content, as well as student exercises, to promote the teaching of the logarithmic relation from its historical roots, which are firmly situated in simultaneous consideration of arithmetic and geometric sequences. Logarithms: The Early History of a Familiar Function - Introduction Author(s): Kathleen M. Clark (The Florida State University) and Clemency Montelle (University of Canterbury) It may come as a surprise to many that often times mathematical concepts don't end up like they started! For those of you who think mathematics is timeless, fixed, and full of unchanging truths, such a proposition may seem unbelievable. But there are many instances in the history of mathematics of the development of a mathematical concept way beyond the purposes and potentialities that its original inventors intended. An example that will be familiar to you all is the logarithm. What is a logarithm? Ask a modern mathematician nowadays and you will get a very different answer from the one you might have got from a mathematician several centuries ago. Indeed, even the very first mathematicians who worked with the logarithmic relation would have given an explanation that would seem quite foreign to a modern mathematician.

So how did the logarithmic relation come about, and how is it that the concept underwent so much change? We will address these questions by looking at the emergence of this concept, and examining some of the issues surrounding its origins. In fact, the question of the origins of the logarithmic relation does not have a simple answer. At least two scholars, the Scottish baron John Napier (1550-1617) and Swiss craftsman Joost Bürgi (1552-1632), produced independently systems that embodied the logarithmic relation and, within years of one another, produced tables for its use. This parallel insight is fascinating and rich in historical detail, and it reveals some methodological challenges for historians of mathematics. In light of all this, we will examine the ideas of these two scholars, as well as explore how historians have portrayed this intricate situation and the questions it raises about mathematics. In large part, we intend to re-introduce teachers to a concept that is often taught without any reference to its original appearance on the mathematical scene.

We hope that a close examination of Napier's and Bürgi's conceptions will enable teachers to consider alternative placement for introducing the idea of logarithms – as part of or after a unit on sequences. Furthermore, we provide in what follows mathematical and historical content, as well as student exercises, to promote the teaching of the logarithmic relation from its historical roots, which are firmly situated in simultaneous consideration of arithmetic and geometric sequences. Logarithms: The Early History of a Familiar Function - Logarithms: A 'Great Tale' for Use in the Classroom Author(s): Kathleen M. Clark (The Florida State University) and Clemency Montelle (University of Canterbury) The logarithmic relation, captured in modern symbolic notation as \[ \log(a\cdot b) = \log(a) + \log(b),\] is useful primarily because of its power to reduce multiplication and division to the less involved operations of addition and subtraction. When this relation hit the scene in the early seventeenth century, its impact was substantial and immediate.

Modern historians of mathematics, John Fauvel and Jan van Maanen ( 2000 ), illustrate this vividly: When the English mathematician Henry Briggs learned in 1616 of the invention of logarithms by John Napier, he determined to travel the four hundred miles north to Edinburgh to meet the discoverer and talk to him in person. (p. xi) Indeed, Fauvel and van Maanen assert that “the meeting of Briggs and Napier is one of the great tales in the history of mathematics” (p. xi). Unfortunately, it seems that many teachers (and their students) are not aware of this particular “great tale” – or, at most, superficially associate the names Briggs and Napier with the invention of the logarithm. Typically, these groups know little about the original conceptions of the logarithmic relation. An anecdote concerning a conversation between Fauvel ( 1995 ) and a colleague reinforces this. Fauvel recounted that when he inquired of his colleague how to teach logarithms, the colleague responded, “Whatever for? Surely no one needs to learn about those any more, now that we have calculators and computers” (p. 39).

Many teachers, when approached about the possibility of teaching logarithms using a historical context, may express the same opinion. Fauvel's counter-argument to his colleague was that “logarithms are a good and accessible example of something fundamentally changing its conceptual role within mathematics” (p. 45). Indeed, examining the historical development of the logarithm with students by exploring arithmetic and geometric progressions allows students “a more deeply rooted understanding of what is going on” (p. 42). The modern concept of the logarithm typically appears late in a second algebra or precalculus course (grades 10 or 11 in the US), situated after a study of polynomial and rational functions, but before sequences and series and conic sections. This organization often leaves students with an impression of disconnectedness between mathematical topics. Complicating matters further, the logarithm is often presented only briefly in such an algebra or precalculus course, in order to lead to a broader study of logarithmic functions to match students' previous study of other functions (e.g., linear, polynomial, rational). Lastly, in order to focus on the study of logarithmic functions, instruction and curricular organization dictate that this function exist as the inverse of the exponential function.

This, in particular, contrasts starkly with the historical circumstances: in fact, the trajectory of modern mathematical pedagogy does not imitate history, as exponentials arrived on the mathematical scene well after the introduction of the logarithm! Victor Katz ( 1995; 1997 ) provided a succinct argument for examining the development of logarithms from a historical perspective. He observed that Napier developed logarithms “for use in the extensive plane and spherical trigonometrical calculations necessary for astronomy” ( Katz, 1995 , p. 49). Although the motivation for developing logarithms is significant, Katz noted that, in general, students today often know very little about astronomy and about the magnitude of both the numbers and the calculations involving such numbers that were necessary to advance the science of astronomy. Astronomical advances have remained critical throughout civilization, however, and Katz ( 1997 ) indicated that, “it is well for us to introduce it [astronomy] whenever possible” (p. 63).

Although there is a strong temptation simply to present the definition and several properties of the logarithm and exercises to practice each, we propose that incorporating original and parallel insights of the logarithm can enrich instruction and learning of the topic, both for this concept and more broadly for a student's understanding of mathematics and its relations and development. Logarithms: The Early History of a Familiar Function - Before Logarithms: The Computational Demands of the Late Sixteenth Century Author(s): Kathleen M. Clark (The Florida State University) and Clemency Montelle (University of Canterbury) The late sixteenth century saw unprecedented development in many scientific fields; notably, observational astronomy, long-distance navigation, and geodesy science, or efforts to measure and represent the earth. These endeavors required much from mathematics. For the most part, their foundation was trigonometry, and trigonometric tables, identities, and related calculation were the subject of intensive enterprise. Typically, trigonometric functions were based on non-unity radii, such as \(R=10,000,000,\) to ensure precise integer output.

* Reducing the calculation burden that resulted from dealing with such large numbers for practitioners in these applied disciplines, and with it, the errors that inevitably crept into the results, became a prime objective for mathematicians. As a result, much energy and scholarly effort were directed towards the art of computation. Accordingly, techniques that could bypass lengthy processes, such as long multiplications or divisions, were explored. Of particular interest were those that replaced these processes with equivalent additions and subtractions. One method originating in the late sixteenth century that was used extensively to save computation was the technique called prosthaphaeresis, a compound constructed from the Greek terms prosthesis (addition) and aphaeresis (subtraction).

This relation transformed long multiplications and divisions into additions and subtractions via trigonometric identities, such as: \[2\cos(A)\cos(B)=\cos(A+B)+\cos(A-B).\] When one needed the product of two numbers \(x\) and \(y,\) for example, trigonometric tables would be consulted to find \(A\) and \(B\) such that: \[x=\cos(A)\;\;{\rm and}\;\;y=\cos(B).\] With \(A\) and \(B\) determined, \(\cos(A+B)\) and \(\cos(A-B)\) could be read from the table and half of the sum taken to find the original product in question. Thus the long multiplication of two numbers could be replaced by table look-up, addition, and halving. Such rules were recognized as early as the beginning of the sixteenth century by Johannes Werner in 1510, but their application specifically for multiplication first appeared in print in 1588 in a work by Nicolai Reymers Ursus ( Thoren, 1988 ).

Christopher Clavius extended the methods of prosthaphaeresis, of which examples can be found in his 1593 Astrolabium ( Smith, 1959 , p. 455). Finally, with the scientific community focused on developing more powerful computational methods, the desire to capture symbolically essential mathematical ideas behind these developments was also growing. In the fifteenth and sixteenth centuries, mathematicians such as Nicolas Chuquet (c. 1430–1487) and Michael Stifel (c. 1487–1567) turned their attention to the relationship between arithmetic and geometric sequences while working to construct notation to express an exponential relationship. The focus on mathematical symbolism in centuries prior and the growing attention to notation–particularly the experimentation with different versions of exponent notation–played a critical role in the recognition and clarification of such a relationship. Now the mathematical connection between a geometric and an arithmetic sequence could be made all the more apparent by symbolically capturing these sequences as successive exponential powers of a given number and the exponents themselves, respectively (see Figure 6 ).

The work on the relationships between sequences was mathematically important per se, but was equally significant for providing the inspiration for the development of the logarithmic relation. ________________________ *Note: Modern trigonometry is essentially based on triangles inscribed in a unit circle; that is, a circle with radius \(R = 1\). Early practitioners used circles with various values for the radius. The relationship between the modern sine function and a sine or half-chord in a circle of radius \(R\) is given by \({\rm Sin}\,{\theta}=R\sin \theta\), where the modern sine function has a lower case '\(\rm{s}\)' and the pre-modern sine an upper case '\(\rm{S}\)'. Logarithms: The Early History of a Familiar Function - John Napier Introduces Logarithms Author(s): Kathleen M. Clark (The Florida State University) and Clemency Montelle (University of Canterbury) In such conditions, it is hardly surprising that many mathematicians were acutely aware of the issues of computation and were dedicated to relieving practitioners of the calculation burden. In particular, the Scottish mathematician John Napier was famous for his devices to assist with computation.

He invented a well-known mathematical artifact, the ingenious numbering rods more quaintly known as “Napier's bones,” that offered mechanical means for facilitating computation. (For additional information on “Napier's bones,” see the article, “ John Napier: His Life, His Logs, and His Bones ” ( 2006 ).) In addition, Napier recognized the potential of the recent developments in mathematics, particularly those of prosthaphaeresis, decimal fractions, and symbolic index arithmetic, to tackle the issue of reducing computation. He appreciated that, for the most part, practitioners who had laborious computations generally did them in the context of trigonometry. Therefore, as well as developing the logarithmic relation, Napier set it in a trigonometric context so it would be even more relevant. Figure 1. John Napier (1550-1617) Logarithms: The Early History of a Familiar Function - Joost Bürgi Introduces Logarithms Author(s): Kathleen M.

Clark (The Florida State University) and Clemency Montelle (University of Canterbury) At about the same time in Switzerland, Joost Bürgi , a court clock maker by profession, grappled with the same issues of computation. Bürgi's key motivation was not only to facilitate computation, but also to produce a single table that could be applied to all arithmetical operations, rather than needing various tables to perform them all. In his work, Arithmetische und Geometrische Progress Tabulen (Arithmetic and Geometric Progression Tables), published in 1620 , Bürgi noted that having separate tables for multiplication, division, square roots, and cube roots is “not alone irksome, but also laborious and cumbersome” (Preface, 1, xi-xii). Figure 5. Joost Bürgi (1552-1632) Logarithms: The Early History of a Familiar Function - The Challenges of Parallel Insights in the History of Mathematics Author(s): Kathleen M.

Clark (The Florida State University) and Clemency Montelle (University of Canterbury) When French mathematician Pierre-Simon Laplace noted that logarithms “by shortening the labors, doubled the life of an astronomer,” ** he hardly exaggerated. Where previously practitioners had been slowed down by long and tedious calculations required in their disciplines, now that logarithms had been introduced, they literally had to spend only half as much time on these necessary computations. Therefore, right from its inception, there has never been any doubt as to the importance of the logarithmic relation. Furthermore, its usefulness has persisted in different areas of mathematics right to the present day. As a result, the logarithm has always formed an important part of historical scholarship, with many historians reflecting on various aspects of its development. In particular, the introduction of the logarithm by (at least) two individuals seemingly simultaneously has led historians of mathematics to wonder to whom to attribute the “discovery.” This issue of priority is a compelling one. When a mathematical idea or concept is introduced independently at roughly the same time by different scholars, historians are faced with an interesting dilemma. They can identify a single individual as ultimately responsible for the idea.

Or, they can resist this inclination to focus on a single person as preeminent over all others, and instead consider on equal footing the multiple proposals given by the various scholars and what these proposals reveal more broadly about the mathematical scene in which these ideas flourished. The former approach often has the consequence of selecting that individual whose proposal most resembles how we understand the concept today, which can often be considered anachronistic and even ahistorical. The latter approach tends to result in a broader contextual historical account and for this reason, many historians would argue, is preferable. One particular historian of mathematics, Florian Cajori , concluded at the very beginning of his work on the history of logarithms, “Few inventors have a clearer title to priority than has Napier to the invention of logarithms” ( Cajori, 1915 , p. 93; for other historians who hold a similar position, see Amy Shell-Gellasch (“ Napier’s e ,” 2008 ) and David E. Smith ( 1959 , p. 149)).

From the outset then, Cajori's focus was on Napier's claim to preeminence with respect to the logarithm concept, and his evaluation consisted of a systematic examination of Napier's contemporaries to reject any claims they might have had to this priority. Indeed, Cajori brings to mind the challenges related to the retrospective appraisal of ideas in history. To find traces of an idea in the work of mathematicians past is too easily done from a future vantage point, and it risks anachronism or ignoring the context in which these ideas were understood. Accordingly, to speak of Napier’s “clear title to priority” to logarithms obscures the fact that his proposal remained always within a particular context for Napier. His logarithm concept was not expansive enough to include the general notion of a function under which we understand logarithms today. Another approach for historians is rather to examine more roundly the various proposals and to refrain from attaching a single historical figure to the emergence of the logarithm. Focusing on the question of priority in mathematics causes scholars to look at the similarities between different manifestations of the same concept.

This is very valuable for the history of mathematics, but emphasizing these similarities can distract from the more insightful issue of how mathematical inventors experimented with the concept, both imperfectly and successfully, and articulated it. Parallel insights reveal an abundant legacy of the motivations of these inventors and of challenges they faced and the ways in which they responded. They reveal the various individual and creative ways in which mathematicians reacted to the intellectual environment around then, and the unique methods or products they developed as a result of these stimuli. Parallel insights are in fact immensely rich for the historian; that legacy is too significant to be obscured by the determination to single out one individual as ultimately preeminent over the others. The parallel insights of Napier and Bürgi epitomize this. As we have seen, both presentations were richly and meaningfully different in many respects. Their conceptions were characterized by distinctive motivations, general designs, presentations, theoretical backdrops, and were also different in many of the details. ________________________ **Note: This quote is frequently invoked in modern scholarship, but its origin remains a mystery to the authors of this paper, who have been unable to track it down in primary source material. Whether or not Laplace actually made the statement remains an outstanding question.

Logarithms: The Early History of a Familiar Function - Parallel Insights and the Reception of Mathematical Ideas Author(s): Kathleen M. Clark (The Florida State University) and Clemency Montelle (University of Canterbury) All of this leads us to the question: what causes some mathematical ideas to get a better reception than others? History shows us that it is not simply a question of mathematical merit, nor the precise instant at which they officially entered the public domain, but that many other factors can be influential. Napier was keen to spread his ideas. We can see this clearly in his endorsement of the translation into English of his work by his contemporary Edward Wright ( 1616 ). Napier expressed his delight in “this secret invention ... so much the better as it shall be the more common.” Evidently, Napier was concerned with disseminating his ideas as widely as possible. In contrast, even Bürgi's contemporaries noted that Bürgi made little effort in promoting his work. Johann Kepler , who was most likely aware of Bürgi's endeavors while the two served in the court of Duke Wilhelm II, the Langrave of Hesse-Kassel (d.

1592), portrayed Bürgi as being neglectful of his ideas and failing to promote them more widely. Kepler noted: These logistic indices ... showed Iustus Byrgi many years before the edition of Napier the way to these very same logarithms. Even so, this man, a procrastinator and guardian of his secrets, abandoned his baby in childbirth and did not nurture it for public use. (As cited in Kepler, Tabulae Rudolphinae, 1627, p. 48 ( Gronau, 1996 , p. 6)) In addition, Napier had two works on logarithms published. Bürgi had only one published, though another was promised. One can only guess at the difference in reception had this second, more “theoretical” work made its way to public consumption. Napier's first work was published in 1614 and was typeset and printed. Bürgi's was published in 1620, and although his tables were printed, his original manuscript remains to the present day only in handwritten form.

Napier was mindful of his audience, tailoring his system to those who used arithmetical operations in a trigonometric context. His table entries, as we have seen, were essentially logarithms of sines of uniformly increasing angles. Bürgi's tables were not compiled in this way and needed (at least) an additional step to be useful in a trigonometric context. Napier was part of the busier academic community, and had his work translated into the vernacular by Wright almost immediately. Bürgi wrote in German, which was arguably less accessible to the wider community. Furthermore, Napier seems to have responded more quickly to the reaction of his contemporaries and to suggestions they had for improving the system. Napier updated his system and made various accommodations so that it would be easier to use. His contemporaries, chief among them Briggs, helped figure out and initiate these changes, and published the results ( Jagger, 2003 ). A primary improvement of Napier's logarithms was the modification that (log 1 = 0), usually credited to Henry Briggs . Most likely this important change was not solely Briggs' idea.

However, the very quick meeting and collaboration between the two men enabled “the idea [to come] about in discussions between Napier and Briggs” ( O'Connor & Robertson, 1999 ). Napier was an established scholar; Bürgi was an employee of the court with (presumably) limited resources and connections, despite the presence of Kepler. Logarithms: The Early History of a Familiar Function - Conclusion Author(s): Kathleen M. Clark (The Florida State University) and Clemency Montelle (University of Canterbury) Logarithms represented at this time in so many ways both what was old and what was new. This relation looked back to reflect concerns of computation, but looked forward to nascent notions about mathematical functions. Although logarithms were primarily a tool for facilitating computation, they were but another of the crucial insights that directed the attention of mathematical scholars towards more abstract organizing notions. But one thing is very clear: the concept of logarithm as we understand it today as a function is quite different in many respects from how it was originally conceived. But eventually, through the work, consideration, and development of many mathematicians, the logarithm became far more than a useful way to compute with large unwieldy numbers.

It became a mathematical relation and function in its own right. In time, the logarithm evolved from a labor saving device to become one of the core functions in mathematics. Today, it has been extended to negative and complex numbers and it is vital in many modern branches of mathematics. It has an important role in group theory and is key to calculus, with its straightforward derivatives and its appearance in the solutions to various integrals. Logarithms form the basis of the Richter scale and the measure of pH, and they characterize the music intervals in the octave, to name but a few applications. Ironically, the logarithm still serves as a labor saving device of sorts, but not for the benefit of human effort! It is often used by computers to approximate certain operations that would be too costly, in terms of computer power, to evaluate directly, particularly those of the form \(x^n\). Logarithms: The Early History of a Familiar Function - Appendix: Student Tasks Author(s): Napier's wonderful world of logarithms | Logarithm Napier's wonderful world of logarithms John Napier's famous book: "The construction of the wonderful world of logarithms".

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Question: What mathematical tabular system was invented in 1614 by John Napier that changes calculations of, for example, multiplication and finding square roots to addition and division respectively?
Answer: Logarithms