Visual Calculus?

Just came across a really interesting follow up to my recent Calculus without Limits blog..

To tempt your interest, here is a problem:

A bicycle rider is riding in a perfect circle and his wet tires leave two concentric circles on the pavement. (if that is difficult to visualize, see here) What is the area between the outer circle and the inner circle.(the area of the annulus). You are allowed to ask for one measurement (other than the answer) that will allow you to solve the problem. What is that measure.

I came across this little gem following up on some information from a blog at Arjen Dijksman’s Physics Intuitions.
Following up on that and searching around led to a truly interesting MAA article by Tom Apostol about a really interesting approach to a “visual calculus” by Mamikon Mnatsakanian. A really great read for calculus teachers, and students.

Who'ddd

Glancing through an old MAA Mathematics Magazine in a list of prominent contributors to the early science of analytic geometry was the name the Dutch politician, Johann Hudde.....

Johann WHO, dude? How do they list a guy with Pascal and Fermat and Descartes and I never heard the name?

It appears that the rise of Leibniz/Newtons calculus based on limits essentially obliterated the record (for most of us) of a brief period when the early calculus flourished with a method that was based on analytic geometry and no Limits...that’s right, calculus without limits.

I searched for Hoode and was hooked when I found a quote that described a comment about him by Leibniz

Leibniz in particular was impressed with Hudde’s work, and when Johann Bernoulli proposed the brachistochrone problem, Leibniz lamented:
If Huygens lived and was healthy, the man would rest, except to solve your problem. Now there is no one to expect a quick solution from, except for the Marquis de l’Hˆopital, your brother [Jacob Bernoulli], and Newton, and to this list we might add Hudde, the Mayor of Amsterdam, except that some time ago he put aside these pursuits .

When I found the whole article, it turned out to be an interesting historical, mathematical journal entry, "The Lost Calculus (1637-1670), Tangency and Optimization without Limits", by Jeff Suzuki in the MAA Mathematics Magazine, Dec, 2005.


In A History of Mathematics, by Cajori, we find that Hudde was the first to use three variables in analytic geometry.

The article by Suzuki would seem to be a wonderful historical read for calculus teachers who, like me, never heard of Hudde’s rule.

Time, and Trig, and Conics, Oh MY!

A while back I wrote about the equation of time, recalling in part,

It came from a lesson in trig on simple harmonic motion. We were talking about things that demonstrated sinusoidal behavior, and one bright young man suggested that the height of the sun at noon would be an example. I sort of agreed with a comment about "not exactly at noon.. but" and then the little guy was confused.. "You know, I said, like today ."(it was Feb 12) "I think the sun was about 12 minutes late or so."
Slow looks at each other, then back to me... the three letter word look,,,,,"Huh?"
"You know, that's what the analemma is for, telling if the sun is early or late.".....

Same look, compounded by the wild eye..."HUH?"

See the whole blog here

So anyway, today I opened the blog over at "Square Circle Z" by Zac,(who seems to prefer to be called, "the mysterious Zac") and it showed a really nice use of the addition of two trig functions to explain the two components of the Equation of time. A nice exercise for people who tend to patienly explain to students who ask "What's that good for?"...(my personal first instinct is to throw things, but many administrations frown on that)

Can Statistics Really be 120 Years Ahead of Science

Scientists have beeen able to use analysis of the 54 Genes that are known to be associated with height to predict the height of a subject. Unfortunatly the new and scientific approach is far less accurate than the statistical model created by Francis Galton 120 years earlier.

The study appears in the European Journal of Human Genetics (2009) 17, 1070–1075;Predicting human height by Victorian and genomic methods Here is the abstract,

"In the Victorian era, Sir Francis Galton showed that ‘when dealing with the transmission of stature from parents to children, the average height of the two parents, … is all we need care to know about them’ (1886). One hundred and twenty-two years after Galton's work was published, 54 loci showing strong statistical evidence for association to human height were described, providing us with potential genomic means of human height prediction. In a population-based study of 5748 people, we find that a 54-loci genomic profile explained 4–6% of the sex- and age-adjusted height variance, and had limited ability to discriminate tall/short people, as characterized by the area under the receiver-operating characteristic curve (AUC). In a family-based study of 550 people, with both parents having height measurements, we find that the Galtonian mid-parental prediction method explained 40% of the sex- and age-adjusted height variance, and showed high discriminative accuracy. We have also explored how much variance a genomic profile should explain to reach certain AUC values. For highly heritable traits such as height, we conclude that in applications in which parental phenotypic information is available (eg, medicine), the Victorian Galton's method will long stay unsurpassed, in terms of both discriminative accuracy and costs. For less heritable traits, and in situations in which parental information is not available (eg, forensics), genomic methods may provide an alternative, given that the variants determining an essential proportion of the trait's variation can be identified. "

From the details of the work, "In a population-based study of 5748 people, we find that a 54-loci genomic profile explained 4–6% of the sex- and age-adjusted height variance, and had limited ability to discriminate tall/short people, as characterized by the area under the receiver-operating characteristic curve (AUC). In a family-based study of 550 people, with both parents having height measurements, we find that the Galtonian mid-parental prediction method explained 40% of the sex- and age-adjusted height variance".

Galton's approach was not "just" the average of the parents heights, but involved the deviation of the midparent (the average of the parents when the female was scaled up by 1.08) from the average midparent. The offspring, Galton determined, would be only 2/3 as far away from the mean as their mid-parent, on average. It is this "regression toward mediocrity" that begat our present term for regression. In his words, "We can define the law of regression very briefly. It is that the height-deviate of the offspring is, on the average, two thirds of the height-deviate of its mid-parentage." [from :Regression towards mediocrity; (1885), p. 252]