My research paper is on John Napier and the invention of logarithms. I know that his motivation for inventing logs was to simplify some of the trigonometric calculations being done in astronomy, but while everyone says that, I can find no example of such a problem. What kinds of problems were astronomers solving? What calculations were they doing circa 1595? And how the hell did they even do trigonometry the way they did, where the "sine" of a 60 degree angle on a circle of radius=100 is entirely different from the "sine" of a 60 degree angle on a circle of radius=1000?

It seems silly to try to write a paper on a topic and not be able to have a single example problem, but that is the quandary in which I find myself.

## 9 comments:

The sine example makes the opposite of sense. Good luck.

A cursory look at the Descriptio, his son's Constructio, and Mark Napier's De Arte Logistca shows that John Napier wrote in the manner of explaining his development of logarithms as a purely mathematical exercise. In none of the manuscripts do the authors relate their work to actual world problems. I believe this is normal for treatises of this type at that time in scientific development. The author explained his development and left it to the reader to understand how to apply it to real world problems.

In fact, in reading about Napier's development of "Napier's Bones" (also called Rods), he seemed to be more involved with how to solve complex math problems using simplifying algorithms. I could not find a reference where Napier was trying to solve real world problems; he was all about solutions to solving the math regardless of the source of the problem.

Therefore, I do not find it silly not to be able to find practical problems of the day. They probably were never linked together in the literature. For you to show an example problem would be something that Napier may very well have never done in his lifetime.

As to your comment about how they did trigonometry, that one I have no idea. The Descriptio was hard enough to follow after Liber I.

Some possible calculations they might certainly in the 1600's are multiplication, division, and exponentiation. Logarithms simplified multiplication, division, and exponentiation to a much simpler addition, subtraction, and multiplication process. For example: To multiply two large numbers, find their logs from a table then you can easily add their logs. Take the result and use the table in reverse to find the antilog, or product. To cube a number for example, it's easier to multiply the log by 3 then do an antilog to get the result. In addition if you have two data points to compare, find their logs. Let's say one result is different by a factor of 3. This would indicate that they are directly proportional, ie. y=kx^3.

Steve, I've read (a translation of) the Constructio from cover to cover, and have read that it's the better of the two works you referenced. He showed how he developed the logs but not what specific types of problem he was trying to solve. I just think my paper would be a lot more "impactful" if I could have an actual example of the type of trigonometry problem he might solve more easily with logs.

Max, you're right--the way they dealt with sines back then was *not* as the ratio we use today. Also, Napier's logs get *bigger* as the number (sine) gets *smaller*. Still, very cool stuff.

Looking at your research problem from the other end, European astronomers were, in 1595, calculating the orbits of the planets based on their observations and trying to reconcile those calculations with the reigning theories regarding how the universe was structured.

At that point, a geocentric (Earth-centered) model of the universe had held sway for something like 1,500 years. That model, developed by Aristotle (384-322 BC) and refined by Claudius Ptolemy (90-168), had the planets and other heavenly bodies orbiting around the Earth in perfectly circular orbits and at a uniform speed. As observed, however, the planets moved in odd ways that were very difficult to reconcile with this model. And as observations improved and grew more accurate over time, the problem only grew worse.

Part of the solution came with the publication in 1543 of Nicolaus Copernicus’s book “On the Revolutions of the Heavenly Spheres.” He postulated a heliocentric (Sun-centered) model in which the Earth was demoted to simply being one of the planets. What Copernicus did not do, however, was challenge the assumption of perfectly circular orbits and uniform speed. And so, men like Tycho Brahe (1546-1601) and Johannes Kepler (1571-1630) were, in 1595, still calculating away, trying to reconcile the planetary observations with predominant theory. Kepler made the final breakthrough, ultimately discovering that planets do not orbit in perfect circles at all. Instead, they trace out ellipses, with the Sun at one focus (Kepler’s First Law of Planetary Motion). Nor do they move at uniform speed—they speed up when closer to the Sun and slow down when further away (Second Law).

Napier corresponded with both Brahe and Kepler :-)

For not knowing *specific* problems they were solving, I was able to write/type about a page on the difficulties involved in doing the trigonometry of the day. Perhaps, if anyone is interested enough to remind me, I can post the entire manuscript here after I turn it in--in about a month!

Yes, please do post it.

What Peggy said. I'd be very interested to read it.

From Eli Maor's "Story of a Number" ... "Rarely in the history of science has a new idea been received more enthusiastically. Universal praise was bestowed upon its inventor and his invention was quickly adopted by scientists all across Europe and even in faraway China. One of the first to avail himself of logarithms was the astronomer Johannes Kepler, who used them with great success in his elaborate calculations of planetary orbits." Kepler's third law: T^2 ~ Rm^3

Henry Briggs had the idea to have log 1=0 instead of log10^7=0, which made them much more convenient to use ... after the tables were recalculated and retabulated (I forget by whom ... maybe Vlacq?), then everyone who had to use math jumped on them ... especially the bankers.

Then the slide rule 1660-ish.

Then Newton and Liebnitz.

and so on.

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