Napier s Logarithms. Simply put, logarithms are mathematical operations that represent the power to

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1 Napier s Logarithms Paper #1 03/02/2016 Simply put, logarithms are mathematical operations that represent the power to which a specific number, known as the base, is raised to get a another number. Logarithms are not only essential to the advanced study of mathematics, but they also play a major role throughout science, nature, and art. One reason for this is because logarithms relate multiplicative changes to incremental changes and geometric progressions to arithmetic progressions (Coolman, 2015). In nature, instances where logarithms can be found are in the Richter Scale, mineral hardness and by the logarithmic spiral within the shape of a nautilus shell. Some ways logarithms display themselves in art is by measuring the intensities of sounds or note pitch in music or the spacing between frets on a string instrument. In the mathematical scientific world logarithms help to explain numerous scientific phenomenons such as, a ph level and the intensity of the brightness of stars (Coolman, 2015). All of these examples provided above prove evidence to the significance that logarithms have in our lives. The person who is accredited with the discovery of logarithms is John Napier, a scottish mathematician. He was seen as revolutionizing the world of mathematics when he published his table of logarithms in his book, Mirifici logarithmorum canonis descriptio (translated as, A Description of the Wonderful Table of Logarithms), in At the time, there was no abstraction that existed about such a relationship, so Napier s discovery of the logarithm was even more impressive because his, conception of a logarithm involved a perfectly clear apprehension of the nature and consequences of a

2 certain functional relationship (Hobson, p.7, 1614). Napier s invention of logarithms provided mathematicians with a tool that would save them much time and effort in solving extensive mathematical calculations. The term logarithms originated with Napier, which he derived from the Greek terms of logos and arithmos, the former meaning proportion and the latter meaning number (Clark and Montelle, 2011). Napier developed logarithms because he wanted to save computational time when using, extensive plane and spherical trigonometrical calculations necessary for astronomy (Katz, p.49, 1995). The table that Napier created was not just a table of logarithms of numbers, but it was a table of logarithms of values of sine functions. At the time, it is important to take into consideration that the cosine and tangent functions had not yet been discovered. Also, the sine function during this time period was thought to be the lengths of specific lines within a circle of large radius (Katz, 1995). Suppose we have the circle pictured below, with radius r and angle θ. Circle Figure 1. (Ayoub, 1993) When Napier was constructing this table he was looking to create a table consisting of sin θ Napier established that the total sine or the log sin 90 = log( 10 7 ), in other words the radius, or r, equaled 10,000,000. Also, this meant that when s in 0 = 0.

3 When Napier developed his theory of logarithms, he offered a qualitative definition that was grounded in a kinetic framework. When looking to find the definition of the logarithmic function, he compared the correspondence between a set of geometric progression and one in the arithmetic progression. The foundation of his model to find the correspondence between the two was rooted in the idea that, the displacement of a point which moves with constant velocity is arithmetic while the displacement of a point which moves with a velocity proportional to the displacement is geometric (Ayoub, p.354, 1993). Below is an example of Napier s model where TS is the fixed length w = 10 7, because he was looking for the logarithms of sin θ, as stated earlier. Also, let line segment OL be an infinitely extending line. Figure 1. (Ayoub, 1993) When Napier was creating this model he had imagined these lines to represent the two parallel paths along which two particles were traveling, where the first line is a fixed length ( TS) and the second line is an infinite length ( OL). He imagined that the two particles would start at the same spots on their respective lines, T and O, when the time was zero or when t = 0. They would also start with the same initial velocity, v 0. He

4 determined that the first particle, point P, on the finite line, TS, would be moving so that, its velocity was proportional to the distance remaining from the particle to the fixed terminal point of the line segment ( Clark and Montelle, 2011). Therefore, the velocity of P is decreasing from its initial velocity to a velocity of zero by the time it reaches point S. In contrast, he determined that the second particle, point Q, traveling on the infinite line of OL would be in uniform motion or would have a constant velocity, so that it was covering equal distances in the same amount of time ( Clark and Montelle, 2011). At any moment, when the distance on the first line, TS, is not covered it was the sine and the distanced spanned on the line OL was the logarithm of sine. Therefore, as the sines decreased in geometric proportion, the logarithms increased in arithmetic proportion. At any given time t, the point P is considered to be distance x from S, whereas for the second line, at any time t, point Q is distance y from the starting point O, which summarizes to Napier s definition of logarithmic concept as y = l ogarithm of x or written as y = L N(x) to distinguish from the modern concept for logarithms. Seemingly, L N(w) = 0, because when P is at point T, it is evident that P S = w while, O Q = 0 (Ayoub, 1993). To transcribe the relationship between the two lines and how they correlate to logs and sines, Napier generated numerical entries to plug into a table. To compute the entries seen below in table 1., he first generated the logarithms and then chose the sine of the arc that corresponded with those logarithms. Table 1. is a reproduction of one of Napier s tables, and to calculate the values in the first column of the table, he would

5 have to use the relation of p n+1 = p n (1 1 ) where p. As shown in the table below, = 10 7 Napier only computed the logarithms for the angles of sine from zero degrees to 90 degrees at intervals of one minute. Table 1. A Reproduction of Napier s Original Table ( Clark and Montelle, 2011). The numbers in bold in the first column, the logarithms in column 2 and the corresponding Sines of the minutes of the arcs were extracted from Napier s original table and then presented in the above table. To see how these values correspond to the numbers in Napier s original table refer to table 2. and look at the final three columns and rows one through six. To arrange these tables he first selected increments of arc θ (Circle Figure 1.) per each minute. Next he listed the corresponding sine for each minute

6 of arc and then, finally, matched it to the corresponding logarithm. In this particular table (table 2.), Napier provides the first half of the first degree in minutes of arc and by using symmetry, he also gives the last half of the eighty ninth degree in minutes of arc. It is noted that Napier himself recalled that computing all of the entries in all three of his tables took him over a span of twenty years and it is estimated that he must have computed over ten million entries all of which he had to determine the appropriate values ( Clark and Montelle, 2011).

7 Table 2. Napier s first table, an excerpt from the first page of tables in his work Mirifici logarithmorum canonis descriptio ( Clark and Montelle, 2011). (Image used courtesy of Landmarks of Science Series, NewsBank Readex)

8 John Napier himself knew the exhaustion of calculating laborious computations stated in his work, Mirifici logarithmorum canonis descriptio (1614) that, Seeing there is nothing that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers,... I began therefore to consider in my mind by what certain and ready art I might remove those hindrances (from the preface of the first English translation in 1616). His determination to tackle the issue of reducing the time it took to make computations lead him to his discovery of the logarithms. Today, we take logarithms for granted, seeing them mostly as the inverse of calculating exponential functions. We often learn how to raise a number to a power before even learning what a logarithm is. However, in a time when there were no such things as calculators, neither mechanical or electrical, Napier s discovery of the logarithm and the construction of his tables revolutionized the world of mathematics. Logarithms were extremely vital in the discoveries that advanced the mathematical and scientific world such as, simplifying computations found in surveying, astronomy, navigation, and later engineering. Napier and his discoveries amplified the way that practitioners calculated computations and helped to advance our world through creating a more simplified way to compute large and complex mathematical equations.

9 Works Cited Ayoub, Raymond. "What Is a Napierian Logarithm?" The American Mathematical Monthly (1993): JSTOR [JSTOR]. Web. 20 Feb Clark, Kathleen M., and Clamency Montelle. "Logarithms: The Early History of a Familiar Function John Napier Introduces Logarithms." Convergence. Mathematical Association of America, Jan Web. 04 Feb Coolman, By Robert. "What Are Logarithms?" LiveScience. TechMedia Network, 22 May Web. 04 Mar Hobson, Ernest William. Cambridge University Press. Cambridge: Cambridge UP, Print. Katz, Victor J. "Napier's Logarithms Adapted for Today's Classroom." Learn from the Masters. N.p.: Mathamatical Association of America, Print. Napier, John. Mirifici Logarithmorum Canonis Descriptio Eiusque Usus, in Utraque Trigonometria ; Ut Etiam in Omni Logistica Mathematica, Amplissimi, Facillimi, & Expeditissimi Explicatio. Edinburgi: Ex Officinâ A. Hart, Print.

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