The Logarithms, Its Discovery and Development
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|Two Pages from John Napier's Logarithmic Table|
You should have realized that multiplication and division done in the geometric series can be translated into addition and subtraction in the arithmetic series. For example, we want to calculate . We go up to find the number that corresponds to 4 and that corresponds to 16 on the first row of the table above, which are 2 and 4, and add them together, which gives up 6, and then come down to find the answer directly below, which is 64. This operation is a direct result of which JN did not explicitly know. Instead, he ingeniously defined logarithms in terms of the continuous motions of two particles. Below is how he defined it originally. He presented his argument in separate articles.
Article 23. To increase arithmetically is, in equal times, to be augmented by a quantity always the same. Article 24. To decrease geometrically is this, that in equal times, first the whole quantity then each of its successive remainders is diminished , always by a like proportional part. Article 25. Whence a geometrically moving point approaching a fixed one has its velocities proportionate to its distances from the fixed one. Article 26. The logarithms of a given sine is that number which has increased arithmetically with the same velocity throughout as that with which radius began to decrease geometrically, and in the same time as radius has decreased to the given sine.
Article 23 means that if we have a point moving with constant velocity, , then in equal time interval, , it will move equal amount of distance. Refer to the diagram above. and hence, the lengths form an arithmetic series.
Article 24 means that if we have a distance of and a point moving in such a way that in equal time interval, , it will move a distance that is of a fixed portion of the distance the point has yet to travel. Say we choose that constant portion as and refer to the diagram above. . As a result, the lengths form a decreasing geometric series.
Article 25 needs a leap in the understanding. JN claimed that the ratio between successive velocities of the particle at are the same as the successive distances that the particle has yet to travel, i.e. . How so? This is JN's argument.
For we observe that a moving point is declared more or less swift, according as it is seen to be borne over a greater or less space in equal times. Hence the ratio of the spaces traversed is necessarily the same as that of the velocities. But the ratio of the spaces traversed in equal times, T1,12,23,34,45,&c (we used TG,GH,HI,IJ), is that of the distances TS,1S,2S,3S,4S,&c(we used TS,GS,HS,IS,JS). Hence it follows that the ratio to one another of the distances of G (the particle) from S, namely TS,1S,2S,3S,4S,&c., is the same as that of the velocities of G at the points T,1,2,3,4,&c.,respectively.
In notation language, he means that
Eqt. 2 is an easy argument to buy. Using the results from Article 24, both sides of Eqt. 2 equals . 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 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?
Article 26 gives the definition of Napier's logarithm. Say we have two particles moving in the ways stipulated by Article 23 and Article 24. At point and point , the two particles have the same velocities. In a certain time period, particle has moved to point and has moved to point . Hence . As a result, which is the result from Article 27. Using calculus, if , we have , where denotes the velocities of and at and ; and if , ; thus . Hence, where is a constant. I have questions about this. I got is from a book.
The limits of a logarithm
Having defined what logarithm is, JN still have no method to approximate with accuracy the values them. However, with his great intuition and ingenuity, he proposed that,
Article 28. Whence also it follows that the logarithms of any given sine is greater than the difference between radius and the given sine, and less than the difference between radius and the quantity which exceeds it in the ratio of radius to the given sine. And these differences are therefore called the limits of the logarithms.
It is indeed a mouthful. This is how he thought about it. If the two particles have the same velocities at point and , then a certain moment later, say seconds, then since moves with constant velocity while is subjected to deceleration. What about seconds previously? Well, will be at with . On the other hand, will be at with I know this is obvious but why?
Given , we have by definition. We know that and . Therefore, . On the other hand we know that and as a result . How do we express in terms of and ? We go back to the original relation below.
Now we have the limits that JN proposed . 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 as the radius of and let and where . Now suppose we have a point, , on the left of such that . Furthermore, let there be a point, , on the right of T such that .
Now, two simple conclusions follow which are and . As a result, . Since , then . Then the argument comes full circle as we know that and what are and but and ! In all, .
As of now, we have three important conclusions,
1. The definition of Napier's Logarithms 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, , starting with and each entry below is of the previous entry. Thus we have the matrix below. The ratio of the last entry to the first, , is approximately by binomial expansion
The second table is a 1 by 51 matrix, , with first entry and subsequent ones of the previous ones. The ratio of the last entry to the first,, is approximately by binomial expansion.
The third table is a 21 by 69 matrix, , with . Each subsequent entry in a row is of the previous one. Each subsequent entry in a column is of the previous one. Thus the entry in row and column is . It is clear by now that Matrix is the result of subdividing the difference between and , and Matrix is the result of subdividing the difference between and .
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, , in Table 1, its logarithms lies between which is . The difference between the limits are really "insensible" so Napier took the average of the two and thus . Repeat the process 100 times, we will have the answer.
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 and thus which gives him . 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, , that is near to a number, , of table 2 is found this way: say , find such that then we have
- Relation. 3 with the last number of first column. Then all logarithms are found. . Find the limits of from table 1 and from table 2, adding them together gives the limits of 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 . 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
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.
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- ↑ The MacTutor History of Mathematics archive, 2006
- ↑ Macdonald, 1888, p. xi-xix
- ↑ Hobson, 1914, p. 26
- ↑ Macdonald, 1888, p. 13
- ↑ Macdonald, 1888, p. 13
- ↑ Macdonald, 1888, p. 15-16
- ↑ Macdonald, 1888, p. 34-35
- ↑ Macdonald, 1888, p. 39
- Hobson, E. W. (1914). John Napier and the invention of logarithms, 1614; a lecture. Cambridge: Cambridge University Press.
- Macdonald, W. R. (1888). The Construction of The Wonderful Canon of Logarithms And A Catalogue. Edinburgh and London: William Blackwood and Sons.
- 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: http://www-groups.dcs.st-and.ac.uk/~history/Quotations/Napier.html
Future Directions for this Page
Finish up the last part and do the Henry Briggs part.
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