The Logarithms, Its Discovery and Development

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Two Pages from John Napier's Logarithmic Table
These are two pages from John Napier's original Mirifici Logarithmorum Cannonis Descriptio (The Description of the Wonderful Canon of Logarithms) which started with the following
which translates into


Basic Description



Introduction: Why is John Napier's Discovery so Extraordinary ?

Today, we regard taking logarithms as nothing but the inverse of calculating exponential. i.e. obtaining x=\log_ab knowing a^x=b. We learn how to take powers before learning how to operate with logarithms and how to differentiate before integrate because that is only natural. However, the order (which is best for understanding and learning) in which we learn mathematical knowledge is not necessarily the order in which they were discovered. That is why we still find JN's discovery as wonderful as it was some 396 years ago.

Consider the following questions: What if we don't have a calculator? How would you calculate \log_57? What if you are neither aware of the notion of index as we now accept as a basic algebraic operation nor the exponential notation? Not only that, what if the decimal notation was not even in popular use? Eventually, how would you even come up with the definition and a table of logarithms? Well, it is impossible you would say because what we take to be facts and the basics were even available. It is like trying to calculus before we even know how to add and subtract. Those were the narrowness of the means available to John Napier who predated Isaac Newton and Leibniz (so calculus and subsequently calculation by means of infinite series were not available him as well). While these obstacles in mathematics were daunting enough, those in society were even greater. The period from 16th to early 17th century was full of transformation and turmoil in religion and politics of the British Islands. In spite of all those difficulties, Napier had accomplished what few even in better condition could not have accomplished.




A Definition of Sine We Don't Know


Arithmetic Progression, Geometric Progression and Napier Logarithms


Arithmetic 1 2 3 4 5 6 ... Logarithms
Geometric 2 4 8 16 32 64 ... Antilogarithms







In notation language, he means that

Eqt. 1        ,
Eqt. 2        .
Eqt. 3        .

Eqt. 2 is an easy argument to buy. Using the results from Article 24, both sides of Eqt. 2 equals \frac {8}{7}. Eqt. 1 is an argument that is difficult to understanding. It is correct to say that in equal time interval, a particle with higher velocity will have a larger displacement compared to the particle with lower velocity and hence the ratio of the displacements will be equal to the ratio of the velocities. But the velocity of the particle on the line TS changes constantly and how can we compare then? As a matter of fact, JN did not prove that because calculus was not invented yet and intuitively assumed the validity of his argument. How can we use calculus to do this?


The limits of a logarithm





\frac {NS}{TS} = \frac {TS}{GS}
\frac {NT+TS}{TS} = \frac {TS}{GS}
NT+TS = \frac {(TS)^2}{GS}
NT = \frac {(TS)^2}{GS}-TS

Therefore NT=\frac {(10^7)^2}{x}-10^7

Now we have the limits that JN proposed 10^7-x<Nap \log x< \frac {(10^7)^2}{x}-10^7. Furthermore, JN proposed that,

Article 39. The difference of the logarithms of two sines lies between two limits; the greater limit being to radius as the difference of the sines to the less sine, and the less limit being to radius as the difference of the sines to the greater sine.

Why is that? JN used the same geometric argument. Say we still have TS as the radius of 10^7 and let BS=x and CS=y where x>y. Now suppose we have a point, V, on the left of T such that \frac {TS}{TV}=\frac{CS}{BC} \cdots \cdots Eqt4. Furthermore, let there be a point, A, on the right of T such that \frac {TS}{TA}=\frac {BS}{BC}.

Now, two simple conclusions follow which are \frac {VS}{TS}=\frac {TS}{AS} and \frac {TS}{AS}=\frac {BS}{CS}. As a result, Nap \log CS - Nap \log BS = Nap \log AS - Nap \log TS. Since Nap \log TS = 0, then Nap \log y - Nap \log x = Nap \log AS. Then the argument comes full circle as we know that TA < Nap \log AS < VT and what are TA and VT but TS \frac {BC}{BS} and TS \frac {BC}{CS}! In all, 10^7\frac {x-y}{x}<Nap \log y - Nap \log x<10^7\frac {x-y}{y}.

As of now, we have three important conclusions,

1. The definition of Napier's Logarithms
Relation. 1        
Relation. 2        

Numbers Which We Can Work With

Now we take a break and play with three tables that JN created. The first table is a 1 by 101 matrix, A, starting with 10^7 and each entry below is 1-\frac{1}{10^7} of the previous entry. Thus we have the matrix below. The ratio of the last entry to the first, 10^7, is approximately 1-\frac{1}{10^5} by binomial expansion

Table 1
i=410^7(1-\frac{1}{10^7})^3=9999997.00000029999999 \approx 9999997.0000003
i=510^7(1-\frac{1}{10^7})^4=9999996.000000599999960000001 \approx 9999996.0000006
i=10110^7(1-\frac{1}{10^7})^{100} \approx 9999900.00049505

The second table is a 1 by 51 matrix, B, with first entry 10^7 and subsequent ones 1-\frac{1}{10^5} of the previous ones. The ratio of the last entry to the first,10^7, is 1-\frac {1}{2000} approximately by binomial expansion.

Table 2
i=410^7(1-\frac{1}{10^5})^3=9999700.00299999 \approx 9999700.00300000
i=510^7(1-\frac{1}{10^5})^4=9999600.0059999600001 \approx 9999600.00599996
i=5110^7(1-\frac{1}{10^5})^{50} \approx 9995001.22480404

The third table is a 21 by 69 matrix, C, with C(1,1)=10^7. Each subsequent entry in a row is 1-\frac {1}{100} of the previous one. Each subsequent entry in a column is 1-\frac {1}{2000} of the previous one. Thus the entry in i^{th} row and j^{th} column is 10^7(1-\frac {1}{100})^{j-1}(1-\frac {1}{2000})^{i-1}. It is clear by now that Matrix B is the result of subdividing the difference between C(1,1) and C(2,1), and Matrix A is the result of subdividing the difference between B(1,1) and B(1,2).

Table 3
i=299950009895050 \cdots5046334.45842676
i=\cdots\cdots \cdots \cdots\cdots

The above three tables were generated by Matlab. To see the complete table, download the plaint text Image:Tables.txt here. For Matlab M-script to generate your own table, click here. How can I upload a regular file and link it?

A Matter of Arithmetics

So now JN had the numbers in geometric progression and the two theorems on the limits of logarithms, he had to start making the table. It was an enormous undertaking considering there was no computer available. But he did!

Using Relation. 1, he managed to find the logarithms of the numbers in the Table 1 with sufficient approximation. For example, for the second number, 9999999, in Table 1, its logarithms lies between 10^7-9999999<Nap \log 9999999< \frac {(10^7)^2}{9999999}-10^7 which is 1.0000000 <Nap \log 9999999< 1.0000001. The difference between the limits are really "insensible" so Napier took the average of the two and thus Nap \log 9999999 = 1.00000005. Repeat the process 100 times, we will have the answer.

Nap \log of numbers in Table 1
<Nap \log x<
i=1...10110^7-x \frac {(10^7)^2}{x}-10^7Nap \log x

Next, he proceeded to Table 2. The nearest number to 9999900, the second number of Table 2, in Table 1 is 9999900.00049505 whose limits are 100.00050495 and 100.00000495. Using Relation. 2, he found that Nap \log 9999900.00049505 - Nap \log 9999900 = 0.0004950569156743573 and thus 100.00000495+0.0004950569156743573<Nap \log 9999900.00049505<100.00050495+0.0004950569156743573 which gives him Nap \log 9999900.00049505 = 1.000005000050325. The next logarithm has double the limits of the previous one. Thus using this method, he managed to get all the logarithms of the numbers in the second table.

there is a little problem here. JN used the 7 decimals for calculation for table 1 and Matlab used something different which I cannot change. so small deviation in table 1 to the 8th decimal place and that results in subsequent difference in the limits and logarithms A number, y, that is near to a number, x, of table 2 is found this way: say y<x, find z such that \frac {z}{10^7}=\frac {y}{x} then we have Nap \log z = Nap \log y - Nap \log x

Relation. 3        . Find the limits of Nap \log z from table 1 and Nap \log x from table 2, adding them together gives the limits of Nap \log y from where the mean can be calculated and JN took that as the true value of the logarithm. Then the logarithm of the next number in the column has double the limits of Nap \log y. In this way, logarithms of all the numbers in the first column of table 3 are found. The logarithms of the first number of the second column is found by using Relation. 3 with the last number of first column. Then all logarithms are found.


The new table with all logarithms of the numbers in third table is called the radical table. Now the first two tables can be discarded as they have served the purpose. For a number that is within the limits of the radical table, its logarithm can be obtained using Relation. 2 with a table number that is nearest to it. For a number beyond the limits of the table, there is a way to do it. JN created a table of radios. A number is multiplied by 2 repeated until it is with in limits of the table. Then the new number's logarithm is found as per normal but the original number's logarithm is found by adding that number and the difference.


Back to Square One

What good is all those logarithms? I though he wanted logarithms of sines and cosines? Of course, let's not get distracted into the petty detail so much that we actually forget what we are doing here. Now the argument comes full swing.

What is the point of all these?

Then with those tools, logarithms of all sines between 0 and 90 can be obtained.

Henry Briggs and the Logarithms to the Base 10

I don't have time for this.

Logarithms to the Base e

I don't have time for this.



Teaching Materials


Related Links

Additional Resources


  1. The MacTutor History of Mathematics archive, 2006
  2. Macdonald, 1888, p. xi-xix
  3. Hobson, 1914, p. 26
  4. Macdonald, 1888, p. 13
  5. Macdonald, 1888, p. 13
  6. Macdonald, 1888, p. 15-16
  7. Macdonald, 1888, p. 34-35
  8. Macdonald, 1888, p. 39


  1. Hobson, E. W. (1914). John Napier and the invention of logarithms, 1614; a lecture. Cambridge: Cambridge University Press.
  2. Macdonald, W. R. (1888). The Construction of The Wonderful Canon of Logarithms And A Catalogue. Edinburgh and London: William Blackwood and Sons.
  3. The MacTutor History of Mathematics archive. (2006, February). Quotations by John Napier. Retrieved from The MacTutor History of Mathematics archive, School of Mathematics, University of St Andrews Scotland:

Future Directions for this Page

Finish up the last part and do the Henry Briggs part.


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