The Harmony of the Harmonic Mean

 

Things happen in threes according to the old myth, and in this case it was true. I was doing some research on the early history of a mathematical problem often called the "cistern" problem. You probably know the type; "If one pipe can fill a cistern in 6 hours and another can fill it in four hours, how long would it take both pipes working together." While I was working on that, I got a nice article sent to me on the first proof that the harmonic sequence diverges... and then, I was reading a blog by Dave Marain Math Notationsin which he posed a problem that asked, in its general form, given a square inscribed in a right triangle (with one corner at the right angle of the triangle), what is the length of a side of the square in terms of the legs of the triangle.

So what do all these have in common with each other. dare I say what makes them in "harmony"?.... the answer is Harmony, or at least the mathematical relationship of the harmonic mean.

To the early Greeks, if Nichomachus can be believed, all the means were descriptive of musical relations. Much is often made of the Harmonic Mean in relation to a musical sense, but this may not represent the Greek view. Euclid used the word enarmozein to describe a segment that just fits in a given circle. The word is a form of the word Harmozein which the more competent Greek Scholars tell me means to join or to fit together. Jeff Miller's Web site on the first use of Mathematical terms contains a reference to the very early origin of the harmonic mean, 'A surviving fragment of the work of Archytas of Tarentum (ca. 350 BC) states, 'There are three means in music: one is the arithmetic, the second is the geometric, and the third is the subcontrary, which they call harmonic.' The term harmonic mean was also used by Aristotle. "
My search for the early roots of the cistern problem had taken me back to Heron's Metre'seis around the year fifty of the common era. The problem became a staple in arithmetics and problem books and was used by Alcuin (775) and appears in the Lilavati of Bhaskara (1150). I found the illustration I used on the blog for The First Illustrated Arithmetic a few days ago, from the 1492 arithmetic, Trattato di aritmetica by Filippo Calandri.

The solution to a cistern problem is the harmonic mean of the times taken by each pipe. For example, one problem asks "If one pipe can fill a cistern in three hours, and a second can fill it in five hours, how fast will the two pipes take to fill the cistern if both are opened at once. The solution is given by finding the average rate of fill of the two rates, the harmonic mean of three and five, which is three and three-quarter hours. But as the name "mean" suggest, that's the average rate of the two so working together, they would take one-half the time, one and seven-eighths hours, or about an hour and 53 minutes.

The Harmonic mean is the reciprocal of the mean of the reciprocals of the values, so for values a and b, the harmonic mean is given bywhich for two numbers can be simplified to the more economical
Heron might have been the first recorded example of a cistern problem, but a problem calling on the reader to use the harmonic mean occurs even earlier in the Rhind Mathematical Papyrus, now located in the British Museum, in problem 76. The problem involves making loaves of bread with different qualities, but the solution is still the harmonic mean. (I have learned from David Singmaster's Chronology of Recreational Mathematics that the cistern problem appeared, perhaps 300 years before Heron's use, in China by Chiu Chang Suan Shu (around 150 BC).


The series of terms formed by the reciprocals of the positive integers is a common torment for college students in their first introduction to analysis. The sequencein which each number gets smaller and smaller seems to very slowly approach some upper limit. Even after adding 250,000,000 terms, the sum is still less than twenty, and yet... in the mid 1300's, Nichole d'Oresme showed that it will eventually pass any value you can name. In short, it diverges, slowly, very, very slowly, to infinity. Even when warned, it seems like students want to believe it converges. A well-known anecdote about a teacher trying to get student's to remember that it diverges goes:

"Today I said to the calculus students, “I know, you’re looking at this series and you don’t see what I’m warning you about. You look and it and you think, ‘I trust this series. I would take candy from this series. I would get in a car with this series.’ But I’m going to warn you, this series is out to get you. Always remember: The harmonic series diverges. Never forget it.”<\blockquote>

By the way, each number in the harmonic series is the harmonic mean of the numbers on each side of it (so 1/2 and 1/4 have a harmonic mean of 1/3), and in fact, of any numbers equally spaced away from it such as 1 and 1/5 also have a harmonic mean of 1/3.

And then, I came across that little problem of a square inscribed in a right triangle. If the two legs are a and b, then the sides of the square will have a length equal to the one-half the harmonic mean of a and b .  More generally, a square inscribe in any triangle with one side along a base will have sides equal to one half the harmonic mean of the base and the altitude to that base.  There are lots of other interesting problems that yield to the use of the harmonic mean, and I mean to write again on that collection.

So I guess things do come in threes, unless I come across another one, but whether it comes in threes or fours, it all seems to work together, in perfect harmony.

Many students who struggle with a different puzzle type problem might want to investigate how it too, relates to the harmonic mean, the one where they ask, If you drive to grandmother's house at 60 miles per hour and drive home at 40 miles per hour, what was your average speed for the round trip?  There are dozens more, so just to get a collection, send your favorite problem related to the harmonic mean, and I'll update as they come along. 

More Geometric Solutions to Quadratics

 A few years back I wrote a paper with the tongue-in-cheek title, Twenty Ways to Solve a Quadratic with reference to the song, "Fifty Ways to Leave Your Lover". It included a variety (actually about 20) of approaches, including some graphic approaches, and some history notes on each method.

Recently while looking into the work of Karl George Christian von Staudt (the 147th anniversary of his death is coming up June 1) I found two more graphic methods I had not previously known. I also found an earlier use of one that I had written about in the article above, plus a very early use that I omitted.

Later I changed the name to "Solving Quadratic Equations By analytic and graphic methods; Including several methods you may never have seen." and posted it at Academia.edu.
I will begin with two very early examples and conclude with the von Staudt example.

One of the earliest graphic examples of a solution has to be from Euclid's Elements, in book 2 proposition 11. Euclid's description of the task is to cut a given straight line into two parts so that the rectangle formed by the whole and one of the parts is equal to the square on the second part. If we call the parts of the line b and x, then what we seek is x^2 =(x+b)b or x^2 = bx+b^2.

The construction in the Elements is pretty brief, finding a midpoint and a couple of compass constructions.
To see the image, and Euclid's solution I leave you to the always wonderful web page on the Elements by Professor David Joyce.

When I wrote the paper on solving quadratics, I credited a method (#13 in the list) this way:

13. Real roots by Lill circle. One of the most unusual graphic methods I have ever seen comes from a more general
method of solving algebraic equations first proposed, to my knowledge, by M.E. Lill, in Resolution graphique des équations numériques de tous les degrés..., Nouv. Ann. Math. Ser. 2 6 (1867) 359--362. Lill was supposedly an Artillery Captain, but his method was included in Calcul graphique et nomographie by a more famous French engineer, Maurice d’Ocagne, who called it the “Lill Circle”.

It turns out that before Lill, a young Scottish mathematician named Thomas Carlyle was credited with using the same circle to solve quadratics (To be fair, Lill extended his method to all polynomials, and even complex roots).
In Sir John Leslie's (who died in 1832) "Elements of Geometry" he writes this method "...was suggested to me by Thomas Carlyle, an ingenious young mathematician, and formally my pupil."

Carlyle skipped right to the diameter of the circle in question. Given a quadratic, x²+bx + c=0, plot points at (0,1) and (-b,c) and construct a diameter. The circle with this diameter will intersect the x-axis at the real solutions (if any exist) of the quadratic.
The circle for y=x² +5x - 6 is shown with the actual parabolic function in red.
Students might be challenged to explain why one end of the diameter will always lie on the function.

In 1908 a little book of less than 100 pages titled Graphic Algebra, written by Arthur Schultze, was published by Macmillan & Co. Schultze was a high school math dept. head at the New York High School of Commerce, and an associate Professor at NYU.
The book is free online in several formats.

On page 47 I found a graphic method of solving quadratic equations I had never seen before. The process uses a standard graph of xy=1. [One of the common approaches when calculators and computers did not exist was to alter an equation to the solution of two equations such as a familiar conic and a straight line.]

I will illustrate his approach with the example he uses in his book. To solve the quadratic equation x² + 2x - 8 =0 He first makes the simple step of dividing all terms by x to get \( x+2 - \frac{8}{x} = 0 ( x\neq 0)\)

Now by substitution of y= 1/x (or xy=1) we get x+2-8y=0. So where both of these equations are true, must be a solution to the original equation. Simply picking a couple of convenient points to plot the line x+2-8y = 0 he determines that when y= 0, x = -2; and when y= 1, x = 6. So we graph the equation xy=1 and then plot the points (-2,0) and ((6,1) and hope for an intersection.


The x-coordinates of the two intersections (red) give us the solutions x={-4; 2}.

Schultze's little book also uses the method from my paper (15. Using the graph of y = x^2 and y = -bx – c to find real roots.) as a graphic solution using a simple conic (y=x²)

And for me, the treat of the day was Karl George Christian von Staudt's graphic method of solving quadratics because it is so different from all the others I've ever learned. Staudt was a student at Gottingen and worked with Gauss, who was at the time the director of the observatory, becoming a very good mathematical astronomer in his own right. He began as a high school teacher as well. His book Geometrie der Lage (1847) was a landmark in projective geometry. It was the first work to completely free projective geometry from any metrical basis. In 1857 von Staudt contributed a route to number through geometry called the Algebra of throws, and he made the important discovery that the relation which a conic establishes between poles and polars is really more fundamental than the conic itself, and can be set up independently.
So here is the solution method used by von Staudt. The challenge to teacher (and students) is to see if you can figure out WHY it works (I found this very difficult):
Using the equation x² - 5x + 6 = 0 we begin by constructing a one unit circle centered at (0,1). Then we construct the points \( \frac {c}{-b} , 0)\) and \( \frac{4}{-b},2)\) For b= -5 and c= 6 we get the points \( \frac {6}{5} , 0)\) and \( \frac{4}{5},2)\)
Construct the straight line through these two points (in red in the image).

The line intersects the circle in the points I, and J, and it is not essential to know their coordinates, but J is (1,1)
Now use (2,0) to project each of these points onto the x-axis. The result is the solutions to the equation. In this case x= {2, 3}


I remember reading about the poet Omar Khayyam who developed a number of geometric solutions to cubics.  He had a nice (tricky, for me ) way to solve equations like x^3 + a^2 x = b.  One example like this was in the work of Cardano after he learned/stole Tartaglia's method for cubics, x^3 + 6x = 20.  A little mental work might find the solution since it is an integer, but long before (about 400 years)  Cardano, Omar had a graphic method somewhat based on the Greek idea of completing the square, you might call it completing the cube.

Here is how it worked.  He broke the problem down into the graph of a circle and a parabola.  The center of the circle was at (b/2a^2,0) and the circle passed through (0,0).  Then he set up a parabola with x^2 = ay.  Working out how he came up with that is a pretty big challenge, but fun when you get it.  

For the Cardano equation x^3 + 6x = 20, 6 is a^2, and b=20, so our circle has a center at (20/(2*6), 0), or just (10/3,0).

x^2 = a y can be simp;ified to y = x^2/a, or in this case, y=x^2/(sqrt(6)).  graphing both on geogebra classic we get:

The intersection at x=2 is our real solution for this problem.

The next thing I thought to try was the famous equation that Rafael Bombelli used to get the whole complex number plain in gear.  It was similar, but with a quirk, x^3 = 15x + 4.  The 15x was on the wrong side.  To use Omar K's method above, the 15 would have to be moved tand become negative, but the poet didn't traffic in no crazy negatives.  I didn't have an example of how he had done this kind of cubic, but a moment of inspiration led be to think of what might well have bee his approach.  

Ir we divide all terms by x, we get x^2 = 15 + 4/x.  Since both sides are equal, we can let each of them equal y and so, y= x^2, and y= 15+ 4/x

So I gave them a run on geogebra.

I thought it was interesting that it had solutions at 4 and -4, but I realized it made sense (of course it does, It's math.) when x = +4, 15 + 4/x = 16 and when x = -4, 15 + 4/x = 14.

I found a paper that listed 19 different types of cubics that the Great Poet solved and the method.  I will play with some of these later.

 

The First Illustrated Arithmetic, and Common Long Division

 

I was researching problems related to the harmonic mean (more of which I hope to share in a later blog or blogs) when I came across a note in David E. Smith's "History of Mathematics" (There are actually used copies for a nickel!) about Filippo Calandri's 1492 arithmetic, Trattato di aritmetica. Smith cites it as the first "illustrated" arithmetic, and checking around, David Singmaster seems to agree.
An actual copy is in the Metropolitan Museum of Art in New York, and they have some images from the woodcuts in the book posted here . The cut above was the one of interest to me as it describes a "cistern problem" which was one of the common recreational problems since the First Century, and one of the problems I was researching when I came across this. The book has another first, it seems to have been the first book to publish an example of long division essentially as we now know it.
Here are some additional notes from my web notes on division that pertain to the long division algorithm and five early methods that were used.

..... is the true ancestor of the method most used for long division in schools today, and was called a danda, "by giving". In his Capitalism and Arithmetic, Frank J Swetz gives “The rationale for this term was explained by Cataneo (1546), who noted that during the division process, after each subtraction of partial products, another figure from the dividend is ‘given’ to the remainder.” He also says that the first appearance in print of this method was in an arithmetic book by Calandri in 1491. The method was frequently called “the Italian method” even into the 20th century (Public School Arithmetic, by Baker and Bourne, 1961) although sometimes the term “Italian method” was used to describe a form of long division in which the partial products are omitted by doing the multiplication and subtraction in one step.

The early uses of this method tend to have the divisor on one side of the dividend, and the quotient on the other as the work is finished, as shown in the image below taken from the 1822 "The Common School Arithmetic : prepared for the use of academies and common schools in the United States" by Charles Davies. Swetz suggests that it remained on the right by custom after the galley method gave way to “the Italian method” in the 17th century. It was only the advent of decimal division, he says, and the greater need for alignment of decimal places, that the quotient was moved to above the number to be divided. In a recent Greasham College lecture by Robin Wilson at Barnard's Inn Hall in London, he credited the invention of the modern long division process to Briggs, "The first Gresham Professor of Geometry, in early 1597, was Henry Briggs, who invented the method of long division that we all learnt at school." (I assume he means with the quotient on top.)

I recently found a site called The Algorithm Collection Project. where the authors have tried to collect the long division process as used by different cultures around the world. Very few of the ones I saw actually put the quotient on top as American students are usually taught. In one interesting note, a respondent from Norway showed one method, then explained that s/he had been taught another way, and then demonstrates the common American algorithm, but adds a note that says, “but ‘no one’ is using this algorithm in Norway anymore.” I might point out that the colon, ":" seems to be the division symbol of choice if this sample can be generalized as it was used in Norway, Germany, Italy, and Denmark. The Spanish example uses the obelisk (which surprised me as it was used almost exclusively in English and American textbooks, and even then seldom beyond elementary school), and the other three use a modification of the "a danda" long division process. The method labled "Catalan" is like the "Italian Method" shown above where the partial products are omitted.

Two verses from a poem by James Clerk Maxwell.

   Two verses from a poem by James Clerk Maxwell.  Think deeply on them at your peril, student!

 A vision of a Wrangle,  of a University, of Pedantry, and of Philosophy

Deep St. Mary's bell had sounded,
    And the twelve notes gently rounded
    Endless chimneys that surrounded
        My abode in Trinity.
    (Letter G, Old Court, South Attics),
    I shut up my mathematics,
    That confounded hydrostatics —
        Sink it in the deepest sea!

    In the grate the flickering embers
  Served to show how dull November’s
  Fogs had stamped my torpid members,
      Like a plucked and skinny goose.
  And as I prepared for bed, I
  Asked myself with voice unsteady,
  If of all the stuff I read, I
      Ever made the slightest use.

The rest, I fear you may seek, on the internet, for a peek.  

Imaginary Numbers, Imaginary Constant, ... History and Etymology of Math Terms

 Imaginary Numbers The word imaginary was first applied to the square root of a negative number by Rene Descartes around 1635.  He wrote that although one can imagine every Nth degree equation had N roots, there were no real numbers for some of those imagined roots.  Around 1685 the English mathematician John Wallis wrote, "We have had occasion to make mention of Negative squares and Imaginary roots".  Some mathematicians have suggested the name be changed to avoid the stigma that it seems to create in young students, "Why do we have to learn them if they aren't even real."  (To be honest, in thirty years as an educator, I never heard the question.)  Perhaps the weight of history is too much to support the change.  

Imaginary and real are found in English in 1668 in Philosophical Transactions. The words are found in a review of the book Geometriæ Universalis: "And for the like reason a Cubick Æquation, having three reals roots, can never be reduced to a pure Æquation, which hath but one onely root, for in these Æquations, Reduction shall no wise profit, for as much as 'tis impossible, by aid thereof to change an Imaginary root into a real one, and the Converse." [Google print search by James A. Landau]

As a way of removing the stigma of the name, the American mathematician Arnold Dresden (1882-1954) suggested that imaginary numbers be called normal numbers, because the term "normal" is synonymous with perpendicular, and the y-axis is perpendicular to the x-axis (Kramer, p. 73). The suggestion appears in 1936 in his An Invitation to Mathematics.

Some other terms that have been used to refer to imaginary numbers include "sophistic" (Cardan), "nonsense" (Napier), "inexplicable" (Girard), "incomprehensible" (Huygens), and "impossible" (many authors).

The first person ever to write about employing the square roots of a negative number was Jerome Cardin (1501-1576).  In his Ars Magna (great arts) he posed the problem of dividing ten into two parts whose product if forty.  After pointing out that there could be no solution, he proceeded to solve the two equations, x+y = 10, and xy=40 to get the two solutions, 

5±15−−√$5\pm \sqrt{15}$  .  He then points out that if you add the two solutions, you get ten, and if you multiply them then the product is indeed forty, and concluded by saying that the process was "as subtle as it was useless."

Cardin also left a seed to inspire future work int he mystery of roots of negative numbers. Cardin had published a method of finding soltuons to certain types of cubic functions of the form x3+ax+b${\mathrm{<br>}}^{}$ .  His solution required finding the roots of a derived equation.  For functions in which the value was negative, his method would not work, even if one of the three roots was a known real solution.  
About thirty years later Rafael Bombelli found a way to use the approach to find a root to x3−15x−4${�}^{}$ with the known solution of four.  He went on to develop a set of operations for these roots of negative numbers.  By the eighteenth century the imaginary numbers were being used widely in applied mathematics and then in the nineteenth century mathematicians set about formalizing the imaginary number so that they were mathematically "real."
Some prefer the term "complex number" (a term first coined by Gauss) for combinations of a real and imaginary number like 2 + 3i, and reserve imaginary for numbers that have no real part.

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Imaginary Unit The imaginary number with a magnitude of one used to represent the square root of −−√$\sqrt{-1}$, has been the letter i since it was adopted by Euler in 1777 in a memoir to the St Petersburg Academy, but it was not published until 1794 after his death.  It seemed not to have gained much use until Gauss adopted it in 1801, and began to use it regularly.  The term, imaginary unit, was first created, it seems, by William Rowan Hamilton in writing about quaternions in 1843 to the Royal Irish Academy.  For his three-dimensional algebra of quaternions, Hamilton added two more imaginary constants, j, and k, which were both considered perpendicular to the i, and to each other.  

The term IMAGINARY PART appears in 1748 in A Treatise of Algebra: In Three Parts by Colin Maclaurin. [Google print search by James A. Landau]

Degree, Gradient, Grads, Gons etc....History and Etymology of Math Terms

 Degree is the union of the Latin roots de, down, and gradus, step.  Gradus is actually derived from the Greek word for "to walk" or "go".  Related words with the same root are congress (come together), regress (go back), and of course grade (the step you are on in school, or earned on an evaluation).  Degree as the measure (or step) of an angle dates back at least to the writings of Chaucer who used the word both in his Canterbury Tails (Squires Tale)  in 1386, and his more famous (then) book on the Astrolabe in 1400.

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Degrees, why 360 in a circle I have seen many responses to why we use 360^o in a circle, but the one that most impressed me was by the one below by the late Alexander Bogomolny.  I have copied the entire thing from his response to a question on  geometry discussion site, so here's why there are 360^o in a circle,
"Babylonians used base 60 notation, which is convenient to divide a whole into 2, 3, 4, ... 30 parts.  Early Greeks then probably divided the radius of a circle into 60 parts. Hence, the diameter must have 120 parts. As Pi was known to be close to 3, the circumference would have 360 parts.
This argument may be used to exonerate the Bible (I Kings. 7:23 and II Chronicles, 4:2) which is said to quote 3 as the value of Pi. Not being a geometry manual, the Bible just picked out a simple approximation to Pi to convey the order of magnitude of the measured quantity. ...
Some history of the sexagesimal (base 60) notations appear in D. E. Smith, History of Mathematics, v2, Dover"
DEGREE for angle measure is found in English in about 1386 in Chaucer's Canterbury Tales: "The yonge sonne That in the Ram is foure degrees vp ronne" [OED]. He again used the word in about 1391 in A Treatise on the Astrolabe: "9. Next this fole with the cercle of the daies, that ben figured in manere of degres, that contenen in nombre 365, dividid also with longe strikes fro 5 to 5, and the nombre in augrym writen under that cercle."

GRAD :Gradus is a Latin word equivalent to "degree." 

Gradian/Grade/Grad/Gon  In trigonometry, the gradian, also known as the gon (from the same Greek root for angle that gave us the gon in polygon) is a unit of measurement of an angle, defined as one hundredth of the right angle; in other words, there are 100 gradians in 90 degrees.

  Grad or grade originally referred to one ninetieth of a right angle, but the term is now used primarily to refer to one hundredth of a right angle.  Some early scientific calculators had a key labeled DRG for selecting between degrees, radians, and grads.  The Sharp EL501X2BWH Engineering/Scientific Calculator shown has the key in the top row 2nd from left next to 2nd Function key.

The OED2 shows a use of grade in English in about 1511, referring to one-ninetieth of a right angle.

The OED2 shows a use of grade, meaning one-hundredth of a right angle, in 1801 in Dupré Neolog. Fr. Dict. 127: "Grade .. the grade, or decimal degree of the meridian." (being French and at this period when France had tried to decimalize clocks, calendars, and pretty much everything, it may be where the first use of grad as 1/100 of a degree began.

You may also hear about centesimal degrees.  In the centesimal system, a right angle is divided into 100 centesimal degrees; each centesimal degree, into 100 centesimal minutes; and each centesimal minute into 100 centesimal seconds. (Centesimal degrees are also known as grads , grades , or gon .)

400 degree compass

If you see a sign on a US highway warning about a steep grade, it refers to the tangent of the angle the road makes with the horizontal.

A sign showing a 6% grade will go down (or up) by six feet per hundred feet of horizontal change.  This seems a little confusing since the mile as recorded by your odometer will be slightly more than a mile to get this mile of horizontal change, think Pythagoras.  Or just don't think about it at all since for most highway grades the difference is very small; as you travel one mile in horizontal change, for a vertical decline of 264 feet, you will actually have to travel about 6.5 feet more than a mile.

Conics from Repeating Decimals, an exploration.

  A short While back, one of John D Cooks tweets directed me to an interesting/surprising conic. 

*Wolfram Mathworld

 

I was surprised , in case you didn't notice, each point is two adjacent digits in the fraction  1/7.  The surprise went on to show that taking the digits in consecutive pairs, (14, 28); (42,85) ... would also form an ellipse.  The short post is here if you want to read before continuing.

Being a curious guy, I started trying this for 1/13 = .076923076923.....

It wasn't an Ellipse, but a hyperbola, but still a conic. The equation is \(141x^2 + 134 xy + 9y^2 - 1872x -684y -4387 = 0\)

 Shock and Awe!  Is this some property of repeating decimals I somehow never learned?

Ok how about 1/17?  Its period was 8 digits, so more points to try.  Turns out, it didn't work.  

1/17

Both the ones that worked had six digit periods, maybe that was a factor.  Couldn't find any more small prime with period six, 14 had period six, but it was the same cyclic pattern as 7.  26 has period 6, repeating 384615 after an initial zero. Even twenty-sixths would be the 1/13 cycle, so they would work. 

3/26 cycled the pattern of 1/26, and produced the same hyperbola, but 5/26 had 923076, producing a different hyperbola.  All the other odd/26 (excepting 13/26) produced some cycle of one of these.  

Some (all?) of this is because the repeat pattern of 1/16, preceded by a zero, is 384615 x two is a cyclic permutation of the repeatng sequence of 1/13.

Deeper into the well, Alice, the rabbit, and me.  1/28 had the same cyclic six digits as seven.  1/35 did also. 1/39 had a new six digit period, 0,2,5,6,4,1.  

Oh, Yes.  Can I get an A-men, children.

So now what.  (right)

I realized that 2/13 had a completely different repeat pattern than 1/13; 1, 5, 3, 8, 4, 6.  So I tried it and it worked also. It was a hyperbola also.(below)

2/39th was different from 1 /39, and 3/39 would revert to the conic for 1/13.  

So on to 2/39; 0, 5, 1, 2, 8, 2. (below left)

And that was totally unexpected, and not a conic.

So! What does that lead toward.  Do only period six reciprocals work? More to try that are not six.  Still haven't found a six other than 7, and multiples of 13 that do work. And are there others that work for 2/p and 3/p... when multiples of 13? What happens to 7x13=91, 010989??? I want to try 1/19, it is the next after 1/7 to have a full p-1 period. Have I focused on individual digits too much and not enough on the fact that pairs of two worked also?

Took time to do 1/19, total failure. First 15 points made two ellipses and a degenerate hyperbola. 

So 1/13 in pairs worked as another hyperbola.

Trying 1/17 next.

1/17 in pairs has all 16 points lying  along two parallel lines of an infinite ellipse, equally divided on both lines.  Maybe I am on to something here.  

 (OK, it looks like two straight lines. After I had written and printed this, some years later I was reading this and wondered if I had assumed it was straight from the graph....posh, I must have checked it.....Ooops, I apparently didn't, because when I "re-checked" it suddenly had a bend.  So in fact it seems to be a hyperbola as well.  (Mia Culpa)

Will try 1/19 next, 18 digits.  Gonna' get crowded on the dance floor tonight. 052631578947368421, wow.

The first six points for 1/19 in pairs fell along a pair of intersecting lines, degenerate hyperbola, and then the next four wander off into space are not part of either.  My silver bullet has turned into pewter.   

Going to look at 1/91 both single digits and in pairs as points.  The digits, 010989, are not like either 1/13 or 1/7. 

The singles worked well

1/91 not 1/19

. The doubles produced a very eccentric ellipse.  

So far of the three I have tried with full periods, 1/7 formed conics as both single digit and paired, 1/17 worked only with paired digits, and 1/19 failed both ways .

Is there a pattern I'm missing. Will try 1/23 next, which will give me two each of full period 6n+1 and 6n-1 primes.

The first 9 points would form either an ellipse or a hyperbola depending on the five points selected, bur none contained more than the minimum five points, with four scattered around.

The pairs attempt also had self destructed by nine points.  

So some full period primes work and some don't with single and/or paired digits .  Some primes and some composite numbers with six digit periods. Need to find more six digit period composites that don't just echo 1/7 or 1/13.  1/63 has fresh six digit period, 015873.  1/77 has 012987. , and 1/84 has 190476. 

1/63 forms an ellipse in both one digit and pairs, 1/77 forms hyperbolas in both methods, and 1/84 failed in both methods.  As the old Soma Cube adds used to say, "Soma Do, Soma Don't"

1/7, 1/13, 1/91, At this Point I made up a fraction with six digit period with only one rule, it obeys the rule that numbers in first half (I chose 123) have their nines compliment in second half (876) and plotted the points with single digits (124/1001 as it turns out) and it was a Hyperbola. When I plotted Pairs they were a hyperbola as well.

Some six digit periods did not have the halves with nine compliments.  All the fractions with full periods have this complimentarity, including 1/19. 1/63 has eight compliments between halves, and 1 /84 has no patterns of compliments in the halves.  1/39 had  compliments of six, 2/39 had no complimentary between halves.  so????  Do failures of six digits period just lack complimentary halves?

Next I tried one with 123 and 8-compliments in second half (765) . Another hyperbola with singles and pairs.

Will 7 compliments work; 123654?  Yes for both.

At this point I tried going to negative compliments. Compliments of -2 would give 1,2,3,-3,-2,-1.  For singles it was another hyperbola, and pairs would not work at all as some digits would be mix of positive and negative digits, numbers that would be really imaginary

No idea what all this suggests, but try your own ideas, and do share what you find! I'm making a spread sheet of all the six digit periods I can find and test noting compliments or not.

One more observation.  The numbers that formed ellipses were 1/7, 1/39, 1/91, and 1/63 .  These all are 4n-1 numbers.  The hyperbolas were formed by 1/13, 1/77 .  These are all 4n+1 numbers.  And 1/17, is a 4n+1 number, but it's repeat period is 16 digits.  Just for a kick I took the first three digits of the repeating digits and appended the first three digits from the second half of the repeating digits, and they formed an ellipse.  

If someone else makes more (any?) sense of this, would love a copy of what you find.

Done Right

 Done Right

According to Hoyle, According to Cocker, nach Adam Riese.  There must be one in every language.  But who were they?  

In the US, and England, it’s according to Hoyle.   Some thought the Hoyle in question was Sir Fred Hoyle, who inadvertently coined the term “Big Bang” for the idea of the sudden expansion of a small point of matter/energy into a massive universe in less than a second.  Sir Fred didn’t buy it for a minute, and attacked the idea promoting his own “steady state universe” theory.  1949 The phrase "Big Bang" is created. Shortly after 6:30 am GMT on BBC's The Third Program, Fred Hoyle used the term in describing theories that contrasted with his own "continuous creation" model for the Universe. "...based on a theory that all the matter in the universe was created in one big bang ... ". *Mario Livio, Brilliant Blunders

Fred Hoyle held on to his belief until the discovery of background radiation.  He was a guy who held his ideas strong and hard, but he’s not the source of the expression, according to Hoyle.

That was another Englishmen, Edmond Hoyle.  Edmond Hoyle (1672 – 1769) was an English writer best known for his works on the rules and play of card games. The phrase "according to Hoyle" (meaning "strictly according to the rules") came into the language as a reflection of his generally perceived authority on the subject; since that time, use of the phrase has expanded into general use in situations in which a speaker wishes to indicate an appeal to a putative authority.

So what about this Cocker?   Here is what I found on a site called World Wide Words:

“Something done according to Cocker was done properly, according to established rules or what was considered to be correct.

The etymological story starts in 1678, when John Hawkins published the manuscript of a book which Edward Cocker had left at his death two years earlier. Cocker had been the master of a grammar school in Southwark, across the Thames from the City of London, and Hawkins was his successor in the post.

 The book, after the fashion of the time, had an expansive title — Cocker’s Arithmetick: Being a Plain and familiar Method suitable to the meanest Capacity for the full understanding of that Incomparable Art, as it is now taught by the ablest School-masters in City and Country.

The Arithmetick (like musick and other words it has since lost its final letter) was an enormous success. It had reached its twentieth edition by 1700 and went through more than a hundred altogether. It was widely used to teach basic arithmetic in English schools for well over a century.“  

Benjamin Franklin' autobiography makes mention that he studied another common English translation, Cocker's Arithmetic, after he moved from his home to Pennsylvania, " And now it was that, being on some occasion made asham'd of my ignorance in figures, which I had twice failed in learning when at school, I took Cocker's book of Arithmetick, and went through the whole by myself with great ease.

According to Google N-gram viewer, between 1900 and 1990, the two expressions were about equally used, but seeming to switch in favor every ten years or so.  Then, around 1995, the Cocker phrase seemed to almost disappear, and the Hoyle quote dominated……. But..suddenly around 2016, it’s popularity roared back, and around 2019 they were nearly equal, with a slight edge to Cocker.  

And Nach Adam Riese, well he was a math man too, and even before Cocker.  He was born the year Columbus made his first voyage to what would come to be known as America.  He wrote a number of books in German for arithmetic and algebra, helping to spread the use of variable based mathematics.  His second book ran for 100 editions. And the expression, Das macht nach Adam Riese... (that gives according to Adam Ries) .  Unlike the other two, it is most often used in arithmetic.  A narrower field perhaps, but it has lasted about five hundred years.