The motivation behind this approach is still not well understood by historians of mathematics. Napier imagined two particles traveling along two parallel lines. The first line was of infinite length and the second of a fixed length see Figures 2 and 3. Napier imagined the two particles to start from the same horizontal position at the same time with the same velocity. The first particle he set in uniform motion on the line of infinite length so that it covered equal distances in equal times.
The second particle he set in motion on the finite line segment so that its velocity was proportional to the distance remaining from the particle to the fixed terminal point of the line segment. Figure 2. More specifically, at any moment the distance not yet covered on the second finite line was the sine and the traversed distance on the first infinite line was the logarithm of the sine.
This had the result that as the sines decreased, Napier's logarithms increased. Furthermore, the sines decreased in geometric proportion, and the logarithms increased in arithmetic proportion. We can summarize Napier's explanation as follows Descriptio I, 1 p. Figure 3. The relation between the two lines and the logs and sines.
Napier generated numerical entries for a table embodying this relationship. However in terms of the way he actually computed these entries, he would have in fact worked in the opposite manner, generating the logarithms first and then choosing those that corresponded to a sine of an arc, which accordingly formed the argument.
The values in the first column in bold that corresponded to the Sines of the minutes of arcs third column were extracted, along with their accompanying logarithms column 2 and arranged in the table. The appropriate values from Table 1 can be seen in rows one to six of the last three columns in Figure 4. Logarithms even describe how humans instinctively think about numbers.
Logarithms were invented in the 17th century as a calculation tool by Scottish mathematician John Napier to , who coined the term from the Greek words for ratio logos and number arithmos. Before the invention of mechanical and later electronic calculators, logarithms were extremely important for simplifying computations found in astronomy, navigation, surveying, and later engineering. Logarithms characterize how many times you need to fold a sheet of paper to get 64 layers.
Every time you fold the paper in half, the number of layers doubles. Mathematically speaking, 2 the base multiplied by itself a certain number of times is How many multiplications are necessary?
This question is written as:. A logarithm can be thought of as the inverse of an exponential, so the above equation has the same meaning as:. This means if we fold a piece of paper in half six times, it will have 64 layers.
Understanding that 1 ml of pure alcohol has roughly 10 22 a one followed by 22 zeroes molecules, how many C dilutions will it take until all but one molecule is replaced by water?
Thus, after 11 C dilutions, there will only be one molecule of the original alcohol left. And since is a very large number, the number. Therefore, since. And Napier is today credited with inventing the natural logarithm — without ever having heard of. Marianne Freiberger is Editor of Plus. Logarithms discovered by Napier in were based on sine tables with 0.
Initially the results are nearly equal to the shortfall from 1. It would be a very onerous task to raise these powers from sine 90 degrees down to sine 1 degree, but this would be helped by by sine 75 degrees equalling 0. Without these tables of logarithms there would be no theory from Nicholas Mercator of the area under a symmetrical hyperbola equalling the log of the distance along the x axis, nor of Isaac Newton's reversion of the hyperbola formula to achieve the infinite series for the antilogarithm e.
This year is the th anniversary of Napier's discovery which is not being properly commemorated largely because modern mathematicians have no idea how Napier achieved it. Napier and Regiomontanus before him knew the formulae for constructing sine and cosine tables.
Basically this is sin2u equals 2sinu. This can be converted into sin2u equals 2sinu. Very few modern mathematicians have grasped that sine 75 degrees raised to the power of 10 equals sine 45 degrees.
Napier's angles just below 90 degrees are incorrect. The arc sine of 0. This could explain why nobody seems to have attempted to list the corresponding arc cosines.
We can hardly blame school maths when the experts are so negligent. You are correct, according to my WPs calculator set to double precision. The difference is. I am still wondering how Napier did it. Something like bisection using square roots maybe? If s is 1, then we have an equality where the shortfall is about However if s is about 69, we have the log of sine 75 degrees.
From log sine 75 degrees to log sine 45 degrees, sine 75 degrees raised to the power of 10 is sine 45 degrees. This power of 10 can be split up to measure the intervening logarithms. It is no use expecting modern mathematicians to know anything about this, submitted by Peter L. Napier mentions amounts close to 0.
Hi Peter, Could you please post some images of the above derivations so that it would be clear. I am also interested to know the resources from which you have got this.
If you have any blog of your own let me know. Thank you. Hi could you let me know the resources to study so that I can understand your comment. I have gone through the book invention of napier logarithm by hobson but couldn't get it. Can you explain me how logarithm were invented at the beginning of 16th century on trigonometric functions.
A video tutorial would help a lot for the further generations to learn these inventions. Thanks for this very interesting article. Further to my previous comments, I have come across an interesting approximate relationship which could have helped Napier in constructing his log tables.
One clue as to how Napier constructed his log tables is obscurely contained in paragraph 44 of the Constructio. One crucial question question about Napier's logarithms which needs to be answered is how he arrived at the values for log2 and 0. This could be his first recognition half a logarithm measure the log of the square root. The log symbol seems to be negative integration. The interesting question has arisen as to whether Napier was aware of the base e for his logarithms.
I have recently come to the conclusion that there is far more to the historical origin of logarithms than most mathematicians suspect. Madhava of Sangamagrama applied calculus to tangent formulae which could have inspired Napier to follow a similar path to arrive at the basic formula for log2. Further to my comment of 13 August , the formula I mention of There are the same number of 9s in 0. For even greater accuracy you add on the same number of 9s to 0.
This I think probably explains how Henry Briggs and others soon after were able to calculate logarithms to a much higher accuracy than Napier was able to achieve. Napier's approach to logarithms Napier's major and more lasting invention, that of logarithms, forms a very interesting case study in mathematical development. The basic idea of what logarithms were to achieve is straightforward: to replace the wearisome task of multiplying two numbers by the simpler task of adding together two other numbers.
In a sense this idea had been around for a long time. Since at least 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. Here the top line is a geometric progression, because each term is twice its predecessor; there is a constant ratio between successive terms. The lower line is an arithmetic progression, because each term is one more than its predecessor; there is a constant difference between successive terms.
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. A continuation of these progressions is the subject of a passage in Chuquet's Triparty Read the passage, linked below, now.
Of course, had he wanted to multiply 5 by 9, say, Chuquet would have been stuck. And in The sand-reckoner, long before, Archimedes proved a similar result for any geometric progression. So the idea that addition in an arithmetic series parallels multiplication in a geometric one was not completely unfamiliar.
Nor, indeed, was the notion of reaching the result of a multiplication by means of an addition. For this was quite explicit in trigonometric formulae discovered early in the sixteenth century, such as:.
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