In particular, the Scottish mathematician John Napier was famous for his devices to assist with computation. In addition, Napier recognized the potential of the recent developments in mathematics, particularly those of prosthaphaeresis, decimal fractions, and symbolic index arithmetic, to tackle the issue of reducing computation. He appreciated that, for the most part, practitioners who had laborious computations generally did them in the context of trigonometry.
Therefore, as well as developing the logarithmic relation, Napier set it in a trigonometric context so it would be even more relevant. Figure 1. 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. As well as providing a short overview of the mathematical details, Napier gave technical expression to his concept.
Despite the obvious connection with the existing techniques of prosthaphaeresis and sequences, Napier grounded his conception of the logarithm in a kinematic framework. 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. In the 19th century A.
Most other logarithmic scales have a similar story. That logarithmic scales often come first suggests that they are, in a sense, intuitive. This not only has to do with our perception, but also how we instinctively think about numbers. Though logarithmic scales are troublesome to many if not most math students, they strangely have a lot to do with how we all instinctively thought about numbers as infants. A change from eight ducks to 16 ducks caused activity in the parietal lobe, showing that newborns have an intuition of numbers.
When thinking in terms of ratios and logarithmic scales rather than differences and linear scales , one times three is three, and three times three is nine, so three is in the middle of one and nine. These were of particular utility for simplifying calculations.
This property of making multiplication analogous to addition enables yet another antiquated calculation technique: the slide rule. Two normal linear rulers can be used to add numbers as shown:. Similar to the procedure shown above, two rulers can be used to multiply when printed with logarithmic scales. These markings also match the spacing of frets on the fingerboard of a guitar or ukulele. 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.
This method was much used by astronomers towards the end of the sixteenth century, particularly by the great Danish astronomer Tycho Brahe, who was visited by a young friend of Napier, John Craig, in So Napier was probably aware of these techniques at about the time he started serious work on his own idea, although conceptually it was entirely different. Napier's definition of logarithm is rather interesting.
We shall not pursue all its details, but just enough to see its approach and character. Imagine two points, P and L, each moving along its own line. L travels along its line at constant speed, but P is slowing down.
P and L start from P0 and L0 with the same speed, but thereafter P's speed drops proportionally to the distance it has still to go: at the half-way point between P0 and Q, P is travelling at half the speed they both started with; at the three-quarter point, it is travelling with a quarter of the speed; and so on. So P is never actually going to get to Q, any more than L will arrive at the end of its line, and at any instant the positions of P and L uniquely correspond.
That is, the numbers measuring those distances. Thus the distance L has travelled at any instant is the logarithm of the distance P has yet to go. How does this cohere with the ideas we spoke of earlier?
The point P, however, is slowing down in a geometric progression: its motion was defined so that it was the ratio of successive distances that remained constant in equal time intervals. But we probably want insight into the cause: What average annual growth rate would account for this change?
I start with a thought like this:. By default, I pick the natural logarithm. Radians are similar: they measure angles in terms of the mover. Not yet. They give us a rate as if all the change happened in a single time period.
The change could indeed be a single year of A banker probably cares about the human-friendly, year-over-year difference. We can figure this out by letting the continuous growth run for a year:. The year-over-year gain is 3. From an instant-by-instant basis, a given part of the economy is growing by 3.
In science and engineering, we prefer modeling behavior on an instantaneous basis. Do bacteria colonies replicate on clean human intervals, and do we wait around for an exact doubling? We know the rate was.
Figuring out whether you want the input cause of growth or output result of growth is pretty straightforward. Imagine we have little workers who are building the final growth pattern see the article on exponents :.
Green at the end of the year. But… that worker he was building Mr. Green starts working as well. If Mr. Green first appears at the 6-month mark, he has a half-year to work same annual rate as Mr. Blue and he builds Mr. Of course, Mr. Red ends up being half done, since Mr. Green only has 6 months. What if Mr. Green showed up after 4 months?
A month? A day? A second? If workers begin growing immediately, we get the instant-by-instant curve defined by ex:. We plug that rate into ex to find the final result, with all compounding included. Using Other Bases Switching to another type of logarithm base 10, base 2, etc. Log base e: What was the instantaneous rate followed by each worker?
Log base 2: How many doublings were required? Log base How many 10x-ings were required? Over 30 years, the transistor counts on typical chips went from to 1 billion How would you analyze this? Doubling is easier to think about than 10x-ing. With these assumptions we get:.
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