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Saturday, May 7, 2011

The Description of the Wondrous Canon of Logarithms

Sent in by fellow math history nerd Elizabeth (aka @cheesemonkeysf):

"I was reading Napier's Preface to The Wondrous Canon of Logarithms (in Latin, so sue me) in preparation for using ...  Log War cards this coming week, and I just totally fell in love with the sweet and completely nerdy generosity of his introduction.

So I'm providing this translation.

I agree with the better translators on *most* of the translation, but they leave out some of the little endearments that make it so charming -- and that make me forgive him (somewhat) his rather cumbersome ideas (thank the gods for calculators in our day).

I especially love his commiseration with his "dearest fellow mathematicians" and with "most diligent students of mathematics." The guy had class. :-)"

The Description of the Wondrous Canon of Logarithms, and the use of that which, not only in Trigonometry, but also in all mathematical logistics, is most fully, most easily, and most expeditiously explained.

By the author and inventor John Napier, ___

On the Wondrous Canon of Logarithms

Since nothing in mathematical practice, my dearest fellow mathematicians, is more tiresome than the great delays suffered in the tedium of lengthy multiplications and divisions, the finding of ratios, and in the extraction of square and cube roots– and in which not only is there the time delay to be considered, but also the annoyance of the many slippery errors that can arise: I had therefore been turning over in my mind, by what sure and expeditious art, I might be able to improve upon these said difficulties. In the end after much thought, finally I have found an amazing way of shortening the proceedings, and perhaps the manner in which the method arose will be set out elsewhere: truly, concerning all these matters, there could be nothing more useful than the method that I have found. For all the numbers associated with the multiplications, and divisions of numbers, and with the long arduous tasks of extracting square and cube roots are themselves rejected from the work, and in their place other numbers are substituted, which perform the tasks of these rejected by means of addition, subtraction, and division by two or three only. Since indeed the secret is best made common to all, as all good things are, then it is a pleasant task to set out the method for the public use of mathematicians. Thus, most diligent students of mathematics,
please accept and freely enjoy this work that has been produced through my good will.


By which all the sines, tangents, and secants,
are set out for you by means of great labour and prolixity;
And which this little table of Logarithms, gentle reader,
Gives to you all at once, without great labour.



  1. Serious question (although it may seem testy, I really want to know). Why teach logarithms? I find my students struggle with things that are so much easier to understand and, since we no longer use them with the regularity that we did before calculators became commonplace, why teach them?

  2. Logarithms become really useful once you get deep into calculus. For a student of mathematics - studying the subject for its own sake, on a conventional university course - a solid grounding is crucial.

    An intuitive grasp of logarithms (and exponentials) also gives you new insights into countless areas of science and the world around us. To an extent, you cannot properly understand multiplication until you understand logarithms.

    But enough of their defense. For all their mathematical purity, logarithms are essentially irrelevant as an everyday tool. If the school maths syllabus was designed to equip students for business, science, and design in the 'real world,' then logarithms would not be on the list.

    Personally, I believe there should be a division between maths 'for its own sake,' and the maths we want our scientists and business leaders to learn. The latter course could be much more focused, and hopefully, widely studied.

  3. Understanding logarithms is pretty important for understanding chemistry, I've been told.

    Also, if you're trying to find when something that grows (or decays) exponentially reaches a certain value, you'd use logarithms. Temperature change is an exponential decay (if the surrounding temperature is constant).

    Also the scales for sound and earthquakes are logarithmic. Also...

  4. 95% of the reason I teach logarithms is because of their connection to exponential functions, which are, in my humble opinion, the single most important topic in algebra.

  5. To some degree, I'm with Alex and R. Wright on this. I found logs & exponentials to be incredibly useful once I hit deeper calculus -- and also in making whatever little sense I've been able to make out in my lifetime of the sciences I've tackled. And I believe the relationship between exponentials and logarithms is an amazing training ground for algebra students in understanding and working with the concept of logically equivalent statements (such as log-base-a-of-x-equals-y is logically equivalent to a-to-the-y-equals-x).

    Once I understood what a revolution they were in the history of mathematics -- and how the work of both Napier and Henry Briggs individually and together revolutionized the study of astronomy -- my interest shot up logarithmically (so to speak). I really *am* moved by the intellectual and collegial generosity that motivated their work on developing these techniques. The problems these guys took on were epic in their day. There’s something heroic about the problems they took on and tried to simplify for their colleagues. Briggs' proposal to reform Napier's cumbersome formulations -- and Napier’s willingness to change his expression of the forms of logs -- is an incredible historical event, as well as a model for collaboration in our own day.

    Can you imagine what would happen nowadays if somebody tried that? They’d be sued by the original inventor of the idea for patent infringement and would then have to countersue for patent infringements of her or his own. If you doubt me, read up on how Silicon Valley company law firms have to deal with these kinds of patent infringement, intellectual property battles, and cross-licensing agreements at the cost of millions of (wasted) dollars each year. I left that world and came back to teaching because I could feel my soul draining out through my feet every time I had to gear up for another battle negotiation on this nonsense.

    There’s so much valuable human background in the history of mathematics -- a lot of decisive turning points -- and that’s a lot to motivate the study of mathematics as a tool.

    Call me a humanities person (why not? I do) but once I started hearing about where all this came from, I had a lot more motivation to understand this stuff. And understanding other people's motivations helps me to motivate my own learning, especially whenever I have to take someone else's complicated or difficult idea apart.

    I think we get into dangerous waters when we talk about reducing the math curriculum to those bits that are required for a career in science or engineering. Math is not an a la carte take-out menu. And we cheat students out of the chance to appreciate the power and elegance of inventions such as logarithms.

    Moreover, since we are teaching human beings -- and not training monkeys to punch calculator buttons -- I think the that should not be ignored. Plus logs and exponentials do form a critical part of the foundation for modern investigation and careers. And they give us ground in which to teach many of the habits of mind students will need to become numerate and numerically literate citizens.

    It does not cause indentured servitude or giant opportunity costs for students to study a little bit of logarithms. It is one friggin' unit in one or two years of the high school curriculum. I think we should get over it and accept that they can be useful tools for teaching students how to think and deepen their thinking process.

    And if, along the way, they become more curious about how human beings have helped each other to work their way out of difficult spots at other challenging times in human history, then maybe that will help us build a more collaborative, peaceful, and compassionate society along the way.

    OK, stepping off my soap box now. Sorry if my rant has offended (or bored the crap out of) anyone.

    - Elizabeth (aka @cheesemonkeysf on Twitter)

  6. Dear Kate, you rock!

    I am new to teaching and blogging except on a different continent. This is a useful and entertaining virtual place to be :-) from both perspectives!

  7. As a recently graduated high school student, who is also quite into math, I find logarithms to be incredibly useful.

    Although it may be of passing interest, using logarithms for manual computation (in devices like the slide rule) is incredibly outdated.

    But the concept of a logarithm has tons of applications in a huge variety of fields, including pure mathematics, science, and business (as Alex mentioned).

    They are as universal as their inverse - exponential functions.


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