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:. Thus if you wanted to multiply two sines, or two cosines, together — a very nasty calculation on endlessly fiddly numbers — you could reach the answer through the vastly simpler operation of subtracting or adding two other numbers.
Let the extremes and bee given, and let the meane proportionall be sought: that commonly is found by multiplying the extreames given, one by another, and extracting the square root of the product.
But we finde it earlier thus; We adde the Logarithme of the extreames 0 and , the summe whereof is which we divide by 2 and the quotient shall be the Logar. By which the middle proportionall , and his arch 45 degrees are found as before Book I, 5 p. Kathleen M. Skip to main content. Search form Search. Login Join Give Shops. Halmos - Lester R.
Ford Awards Merten M. Author s :. John Napier from MacTutor History of Mathematics Archive Napier first published his work on l ogarithms in under the title Mirifici logarithmorum canonis descriptio, which translates literally as A Description of the Wonderful Table of Logarithms. Napier's two parallel lines with moving particles Image used courtesy of Landmarks of Science Series, NewsBank-Readex 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.
The following is based on the Constructio , Macdonald's translation , and Goldstine's summary in A History of Numerical Analysis from the 16th through the 19th Century , Properties of geometric and arithmetic progressions were well-known by Napier's time, and the connection between a sequence of powers and its corresponding sequence of exponents that we call the law of exponents has roots in ancient mathematics cf. Euclid IX. I do not know what enabled Napier to make the key connection that led to logarithms or on the other hand, what prevented the discovery earlier.
My take on Napier is that calculating was thought primarily in terms of integers or ratios of integers or fractions , and Napier is at pains to make his calculation of logarithms in terms of integers accurate. His break-through idea is quite ingenious and surprising to me, at least. It is to consider two points moving continuously, one representing the logarithm increasing "arithmetically" while the other representing what we would now call the argument decreasing "geometrically.
Like many first tries, this led to some inconveniences. By the time his table was published, Napier had in mind some improvements, which are included in an appendix to the Constructio. It is large, it appears, so that one can express logarithms accurately in terms of integers.
Briggs too saw some room for improvement, and this led him to seek out Napier. Briggs published a table in Logarithmorum Chilias Prima , the first published table of base logarithms, and contributed some Notes in another appendix to the Constructio. Napier deals with velocities and distances as easily as we deal with derivatives and integrals. In fact, from a modern point of view, there's is no real difference. In our modern terms, we can represent Napier's definition of a logarithm Prop.
The "artificial number" logarithm of a given sine is that which has increased Arithmetically always with the same velocity as the total sine [began] to decrease Geometrically and in the same time as the total sine decreased to that given sine.
Throughout the Constructio , Napier calls the logarithm numerus artificialis and the given number either sinus or numerus naturalis. The "total sine" is a fixed length through which the geometrically moving point can move. It can be translated as "radius" and a given sine will be less than or equal to this; however, trigonometry does not figure in the construction, so the names are somewhat meaningless.
Then by Prop. The notation here is the same as Goldstine's. It is easy to prove these inequalities from properties of the integral, but Napier did in terms of velocities and distances.
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. Aside, this is less than half of the 30 C dilutions common in homeopathy, which shows why the practice is irreconcilable with modern chemistry.
Most scientific calculators only calculate logarithms in base 10, written as log x for common logarithm and base e , written as ln x for natural logarithm the reason why the letters l and n are backwards is lost to history. The number e , which equals about 2.
To do a logarithm in a base other than 10 or e , we employ a property intrinsic to logarithms.
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