A History of the Discovery of Logarithms by Lica Marhao

Purpose and Definition The basic idea of what logarithms were to achieve was straightforward: to replace the wearisome task of multiplying two numbers with many digits by the simpler task of adding together two other numbers. John Napier used 𝑙𝑜𝑔 𝑁 (𝑎) + 𝑙𝑜𝑔 𝑁 (𝑏) to recover the product 𝑎×𝑏 He called these numbers at first ‘artificial numbers’ and later ‘logarithms’. The term “logarithm” was coined from two Greek words meaning something like ‘ratio-number’ (or number of a ratio). John Napier, the mathematician credited with the discoverer of logarithms, when multiplying two numbers 𝑎×𝑏 he assigned to 𝑎 and 𝑏 two other numbers which we will denote as: 𝑙𝑜𝑔 𝑁 (𝑎) and 𝑙𝑜𝑔 𝑁 (𝑏). Napier called these replacements at first ‘artificial numbers’ and later ‘logarithms’. He used their addition to recover the product 𝑎×𝑏.

The Fundamental Property of Logarithms 𝑙𝑜𝑔 𝑁 (𝑎) + 𝑙𝑜𝑔 𝑁 (𝑏) → 𝑎×𝑏 If 𝑓 𝑎+𝑏 =𝑓 𝑎 +𝑓(𝑏) for any 𝑎 and 𝑏 in the domain of 𝑓(𝑥) than 𝑓(𝑥) is a logarithmical function, or a logarithm. The fundamental property of logarithms is that from the sum of two logarithms 𝑙𝑜𝑔 𝑁 (𝑎) and 𝑙𝑜𝑔 𝑁 (𝑏) of two numbers 𝑎 and 𝑏, the product of the numbers 𝑎 and 𝑏 could be recovered. Any function that transforms addition into multiplication must be a logarithm: This property allowed the identification of the function that modeled the area under the curve 1 𝑥 as a logarithm, which we now know as the natural logarithm.

Why a Greek term for this artificially created number? “logarithm”=‘ratio-number’ (or number of a ratio)

From ancient Greek times it had been known that multiplication of terms in a geometric progression could correspond to addition of terms in an arithmetic progression. For instance, consider and notice that the product of 4 and 8 in the top line (32), lies above the sum of 2 and 3 in the bottom line (5). The idea conveyed to us is that addition in an arithmetic series parallels multiplication in a geometric one 𝑎 𝑚 𝑎 𝑛 = 𝑎 𝑛+𝑚 2 4 8 16 32 64 1 3 5 6 Here the top line 2 4 8 16 32 64 is a geometric progression, because each term is twice its predecessor; there is a constant ratio ( 𝑟=2) between successive terms. The lower line 1 2 3 4 5 6 is an arithmetic progression, because each term is one more than its predecessor; there is a constant difference  (𝑑=1) between successive terms. In a mathematical language easily understood by a contemporary algebra student, the exponents of numbers in a geometric sequence are additive when terms are multiplied together:

A historical note: Precisely these two lines 2 4 8 16 32 64 1 3 5 6 appear as parallel columns of numbers on an Old Babylonian tablet, so the idea of creating a correspondence between an arithmetic series and a geometric one may be older than Greek mathematics. Precisely these two lines appear as parallel columns of numbers on an Old Babylonian tablet, though we do not know the scribe's intention in writing them down, so the idea of creating a correspondence between an arithmetic series and a geometric one may be older than Greek mathematics.

cos 𝑥 cos 𝑦 = 1 2 ( cos 𝑥+𝑦 +𝑐𝑜𝑠 𝑥−𝑦 ) Prosthaphaeresis: Obtaining multiplications by means of addition using trigonometry The technique of prosthaphaeresis (the Greek word for addition and subtraction) was discovered by the Egyptian mathematician Ibn al Haytham or Alhazen (known to medieval Europeans as the second Ptolomeus) in 1005, and reintroduced it to the mathematical world by the French mathematician François Viète in the 1580’s It was adopted by the Dane Tycho Brahe for performing astronomical calculations.

The technique of prosthaphaeresis (the Greek word for addition and subtraction) was discovered by the Egyptian mathematician Ibn al Haytham or Alhazen (known to medieval Europeans as the second Ptolomeus) in 1005, and reintroduced it to the mathematical world by the French mathematician François Viète in the 1580’s It was adopted by the Dane Tycho Brahe for performing astronomical calculations.

cos 𝑥 cos 𝑦 = 1 2 ( cos 𝑥+𝑦 +𝑐𝑜𝑠 𝑥−𝑦 ) Tycho Brache used cos 𝑥 cos 𝑦 = 1 2 ( cos 𝑥+𝑦 +𝑐𝑜𝑠 𝑥−𝑦 ) Napier’s idea was conceptually an entirely different technique than using trigonometric formulae, and it was based on a unique approach on interpreting motion. Tycho Brache A young friend of Napier by the name John Craig visited the astronomer Tycho Brache in 1590. Upon his return to England, Craig mentioned the use of the prosthaphaeresis technique cos 𝑥 cos 𝑦 = 1 2 ( cos 𝑥+𝑦 +𝑐𝑜𝑠 𝑥−𝑦 ) to Napier, which in turn motivated Napier to start devoting serious work on a personal idea of how to map multiplications by additions.

A short history of John Napier the author of logarithms: Napier's interest in mathematics, and in computational methods in particular, seems to have started in his early twenties, yet he never attended a university. For many years, John Napier spent his leisure time devising means for making arithmetical calculations easier. The need for better computational methods may have had something to do with the fact that Napier had a reputation for not trusting other people very well. He didn’t trust his servants, (so he ingeniously tested them at times to see if they steal from him), but mostly he didn’t trust accountants and the doctrines and practices of the Catholic church. In his early life he set off to create a table of multiplications for numbers up to ten thousand times ten thousand, that was to be published as an aid for owners of small estates to empower them then check on their accountants if they steal. In the process, he discovered a short cut for multiplying long numbers and managed to develop a method for multiplication which is now known as Napier’s bones.

Multiplication using “Napier’s bones” Napier’s bones consisted of the columns of a multiplication table inscribed on rods, which could make the multiplying of two numbers easier by setting down the partial products more swiftly.  

“Mirifici logarithmorum canonis descriptio" (1614),. Napier's work in mathematics that made him famous over time was first his table of logarithms, “Mirifici logarithmorum canonis descriptio" (1614),. “Mirifici logarithmorum canonis constructio”, published two years after his death in 1619, accounted how logarithms themselves were calculated. Up to the publication of his description of logarithms in 1614, three years before his death, Napier was yet best known to the world for his Protestant religious treatise “A plaine discovery of the whole Revelation of Saint John” (1594), a well-received anti- Catholic work translated into several foreign languages, which argues amongst other things that the pope is the Antichrist. The titles of his well-known works are revealing a contrast not at all surprising given his interests.

“A logarithmic table is a small table by the use of which we can obtain a knowledge of all geometrical dimensions and motions in space...” John Napier Napier’s logarithm Napier originally conceived of his "artificial numbers" or logarithms, (or numbers of the ratio) in purely kinematical terms.

To better understand Napier’s definition let’s look at the following scenario: “Suppose two runners are at some point running at the same speed parallel to each other. After that point, one of them maintains the speed but does not ever stop running, whereas the speed of the other drops in proportion to his distance from a certain finish line he has to reach. The closer this runner gets to the finish line, the slower he runs. Thus, although always moving forward, the slower runner never reaches the finish line, and the constant speed runner never stops, so they both will run forever. “ This is exactly the kind of scenario that Zeno of Elea (a Greek mathematician and philosopher) used to illustrate paradoxes of motion. Zeno argued that it is impossible for a runner (Achilles) to catch up with a tortoise in a race course if the tortoise started the race at an earlier time.

Imagine the two runners are represented by two points, P and L, each moving along its own line.

L travels along its line at constant speed, but P is slowing down   The line segment  𝑃 0 𝑄 is of fixed, finite length, but L's line is endless.  L travels along its line at constant speed, but P is slowing down Hence, Napier’s definition of logarithms: At any instant, the distance  𝐿 0 L is the logarithm of the distance PQ. P and L start (from  𝑃 0  and  𝐿 0 ) with the same speed, but thereafter P's speed drops proportionally to the distance it has still to go: at the half-way point between  𝑃 0 and Q, P is travelling at half the initial speed; at the three-quarter point, it is travelling with a quarter of the initial speed; and so on… So P is never actually going to get to Q, any more than L will arrive at the end of its line (which is infinite), and at any instant the positions of P and L uniquely correspond. Two thousand years after Zeno, Napier’s scenario of the two runners is creating a continuous mapping of the real numbers to his "artificial numbers“ or logarithms.

How does Napier’s definition of a logarithm cohere with the idea that addition in an arithmetic series parallels multiplication in a geometric one earlier? The point L moves in an arithmetic progression: There is a constant difference between the distance L moves in equal time intervals. That is what ‘constant speed’ means. The point P, however, is slowing down in a geometric progression: P’s motion was defined so that it was the ratio of successive distances that remained constant in equal time intervals. The continuous nature of the straight line and motions extends the mapping of a multiplication to addition not only for some particular discrete geometrical progression but for all positive real numbers. Napier’s logarithm applied to a geometric progression (the logarithm of the distance PQ) transforms it to an arithmetic progression (the distance  𝐿 0 L), therefore it transforms a multiplication into an addition. What was clever and revolutionary about Napier’s approach was the use of continuous nature of the straight line and motions which extends the mapping of a multiplication to addition not only for some particular discrete geometrical progression but for all positive real numbers.

The relationship between Napier’s logarithm and other logarithms

The distance  𝐿 0 L as the logarithm of the distance PQ describes a decreasing function. Napier’s logarithm function is consistent with a reflection about the x axis and a vertical stretch/shrink of a logartithmic function as defined today. At any instant, the distance  𝐿 0 L is the logarithm of the distance PQ, thus the distance L has travelled at any instant is the logarithm of the distance P has yet to go. For the original logarithm found by Napier, the shorter the length of 𝑃𝑄 (as P approaches Q), the larger the logarithm. For 𝑥=𝑃𝑄, the smaller the value of 𝑥 the larger the result (or the logarithm), which is the way we would describe a decreasing function. Let us compare the overall behavior of a known logarithm with the above result that holds true for Napier’s logarithm. Provided is a graph of a logarithm: A logarithm is an increasing function, therefore Napier’s logarithm appears to be different than a basic logarithm in use today, yet it would hold true for an x axis reflection of the above graph combined with a stretch/shrink.

With an appropriate choice of units, we can express: the position 𝒙 of the point P as a function of time 𝑥 𝑡 the position "1" representing the finish line for the slower runner (the point Q). the position 𝒚 of the point L as a function of time such that 𝑦(𝑡) = 𝑡 The scenario of the two runners will described by the differential equation 𝑑𝑥 𝑑𝑡 =1−𝑥 The solution of this differential equation is 𝑥 𝑡 =1− 𝑒 −𝑡 with 𝑡=𝑦(𝑡) Short proof: 𝑑𝑥 𝑑𝑡 =1−𝑥 𝑑𝑥 1−𝑥 = 𝑑𝑡 −ln 1−𝑥 =𝑡+𝐶, 1−𝑥>0, 1−𝑥=𝑘 𝑒 −𝑡 , 𝑥 0 =0 𝑥=1− 𝑒 −𝑡 or 𝑥 𝑡 =1− 𝑒 −𝑡 Let us model Napiers’ scenario of the two runners using today’s mathematical notations and knowledge of differential calculus which was invented by Newton and Leibniz 60 years after Napier’s death.

Napier’s base is 𝐍= 𝟏 𝒆 Napier defined 𝑦 𝑡 as "logarithm" of 1− 𝑥 𝑡 or 𝑡= 𝑙𝑜𝑔 𝑁 (1−𝑥 𝑡 ) . Using the solution of the differential equation that modeled the motion 𝑥 𝑡 =1− 𝑒 −𝑡 𝑡= 𝑙𝑜𝑔 𝑁 1−𝑥 𝑡 = 𝑙𝑜𝑔 𝑁 1− 1− 𝑒 −𝑡 = 𝑙𝑜𝑔 𝑁 ( 𝑒 −𝑡 ) Therefore 𝑙𝑜𝑔 𝑁 𝑒 −𝑡 =𝑡 if 𝑁= 𝑒 −1 or 𝑁= 1 𝑒 since 𝑙𝑜𝑔 𝑒 −1 ( 𝑒 −1 ) 𝑡 =𝑡 Using the change of the base formula: 𝑙𝑜𝑔 1 𝑒 𝑥 = 𝑙𝑜𝑔 𝑒 −1 𝑥 = l𝑛(𝑥) ln( 𝑒 −1 ) =−ln(𝑥) Napier’s logarithm 𝑙𝑜𝑔 𝑁 𝑥 is an x axis reflection of ln(𝑥) Therefore Napier’s original definition of a logarithm is related to the natural logarithm with the base 𝑒, where 𝑒 𝑡 is also the function that equals its own derivative.

Napier’s logarithm is changed to base a 10 logarithm by Henry Briggs and renamed as a Briggsian logarithm In 1624 Briggs published Arithmetica Logarithmica, a work containing the logarithms of thirty thousand natural numbers to fourteen decimal places. The base 10 logarithm was the only logarithmic use for 100 years, hence its name ‘common logarithm’ and the lack of base notation when using it for calculations. Henry Briggs In collaboration with Oxford professor Henry Briggs, in 1617 Napier refined his original table of logarithms by constructing tables for logarithms in base 10. The base ten logarithm came down in history as Napier’s logarithm as well as a Briggsian logarithm. Redefining Napier’s logarithm brought some advantages, but it also postponed the discovery of the natural logarithm by more than 100 years

From astronomy to logarithms and then back to astronomy From Tyhco Brahe to Napier, and from Napier to Johannes Kepler

James VI of Scotland (later James I of England and Ireland) Tycho Brahe Napier might never have set aside his anti-Catholic polemics to work on producing his table of logarithms had it not been for the off-hand comment on the method of made prosthaphaeresis by his friend John Craig (Dr. John Craig), who was the physician to James VI of Scotland (later James I of England and Ireland). In 1590 Craig accompanied king James to Norway to meet his prospective bride Anne, who was supposed to have journeyed from Denmark to Scotland the previous year, but had been diverted by a terrible storm and ended up in Norway. On the journey home the royal party visited Tycho Brahe's observatory on the island of Hven, and were entertained by the famous astronomer, well known as the discoverer of the "new star" in the constellation Cassiopeia. During this stay at Brahe's lavish Uraneinborg ("castle in the sky") Dr. Craig observed the technique of prosthaphaeresis that Brahe and his assistants used in their calculation. When he returned to Scotland, Craig mentioned this to his friend the Baron of Murchiston (aka John Napier), and motivated Napier to devote himself to the development of his logarithms and the generation of his tables. During this time Napier occasionally sent preliminary results to Brahe for comment. John Napier James VI of Scotland (later James I of England and Ireland)

Planets move in ellipses with the Sun at one focus. Johannes Kepler Planets move in ellipses with the Sun at one focus. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. III. The proportion between the periodic times of any two planets is precisely one and a half times the proportion of the mean distances. This law was later refrazed as: The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit. Kepler originally used the words “ the proportion” to signify the logarithm of the ratio, so he was asserting that 𝑙𝑜𝑔 𝑇 1 𝑇 2 = 3 2 𝑙𝑜𝑔 𝑟 1 𝑟 2 where 𝑇 1 𝑎𝑛𝑑 𝑇 2 are the orbital periods and 𝑟 1 𝑎𝑛𝑑 𝑟 2 are the mean radii ( or half of the length of the major axis) of the orbits of any two planets. By the year 1605 Johannes Kepler, working with the relativistic/inertial view of the solar system suggested by Copernicus, had already discerned two important mathematical regularities in the orbital motions of the planets: After finding the above two "laws" of planetary motion (first published in 1609) from the observational data of Tycho Brahe, there followed a period of more than twelve years during which Kepler exercised his ample imagination searching for any further patterns or regularities in the data. However, despite all his ingenious efforts during these years, he was unable to discern any significant new pattern beyond the two empirical laws which he had found in 1605 until March 8th 1618 when he suddenly realized that

Kepler’s insight in the form of a diagram: In modern data analysis the log-log plot is a standard format for analyzing physical data. However, logarithmic scales had not yet been invented in 1605. Yet, after twelve years of struggle, this way of viewing the data suddenly "appeared in his head" early in 1618. Kepler initially described his "Third Law" in terms of a 1.5 ratio of proportions, exactly as it would appear in a log-log plot, rather than in the more familiar terms of squared periods and cubed distances. This is no coincidence, since John Napier's "Mirifici Logarithmorum Canonis Descripto" (published in 1614) was first seen by Kepler towards the end of the year 1616. Kepler was immediately enthusiastic about logarithms after reading Napier’s treatise. He even wrote a book of his own on the subject in 1621, the “Rudolphine Tables”. By the 18th of May, 1618, Kepler had fully grasped the logarithmic pattern in the planetary orbits. Kepler’s insight in the form of a diagram:

Kepler announced his Third Law in “Harmonices Mundi,” published in 1619, and also included it in his “Ephemerides” of 1620. The latter was actually dedicated to Napier, who had died in 1617. The cover illustration showed one of Galileo's telescopes, the figure of an elliptical orbit, and an allegorical female crowned with a wreath consisting of the Naperian logarithm of half the radius of a circle. It has usually been supposed that this work was dedicated to Napier in gratitude for the "shortening of the calculations", but most likely because Kepler recognized that it went deeper than this, that the Third Law is purely a logarithmic harmony. A purely mathematical invention, namely logarithms, whose intent was simply to ease the burden of manual arithmetical computations, led directly to the discovery or formulation of Kepler's third law of planetary motion. A purely mathematical invention, namely logarithms, whose intent was simply to ease the burden of manual arithmetical computations, led directly to the discovery or formulation of Kepler's third law of planetary motion.

Lab Time!