Composer of the Day

The Composer of the Day is Percy Grainger, born in a suburb of Melbourne, Australia on 8 July 1882. He was a true eccentric, but an extremely talented eccentric. (If you’re weird but not talented, you’re just “odd”.) Like a few other composers (Rachmaninov), his best-known work, “Country Gardens”, was his least favorite. He hated it, but whenever he gave a concert, people insisted he play it as an encore.

He didn’t like the standard Italian annotations (“molto allegretto” &c), and wrote them in English (“somewhat pertly”). He lived recently enough that you can find recordings of him playing – there’s a YouTube cut of “Irish Tune” (better known today as “Danny Boy”) . The sheet music goes by as he plays. The tempo is marked “Slowish, but not dragged, and in wayward time”. (Either he or the editors put in the Italian notations.) The dynamics run from ff to pppp. It’s lushly harmonized, with the melody in the upper left hand. And made many piano rolls – so you could have Percy playing in your parlour.

A true eccentric, he rated himself the 9th-best composer ever – between Mozart (27 Jan 1756, not quite so good,) and Delius (29 Jan 1862, a little better), with Bach (21 March 1685) – to no-one’s surprise, Nr 1.

He met Grieg (15 June 1843), and performed his piano concerto in concert. During the 1920s, he earned the equivalent of $60,000/week on the concert stage. A bit of an overachiever, he spoke 11 languages fluently (I wonder if there’s a connection between language and music). When he married, it was at the Hollywood Bowl.

He was one of the first to go seriously into electronic music – he wrote a piece called “Free Music No. 1 (For Four Theremins)”, which has recently been adapted for 4 iPhones.

He died in 1961 in White Plains, New York. There is, of course, a Percy Grainger Society.

Circles and Spheres

Standard formulas for the area of a circle, and the surface area and volume of a sphere, involve the constant π. Most of us remember that π is about 3.1416. That’s pretty close, but not exact. Even closer, π is about 3.1415926536. Now that’s really close, but still not exact. π has no exact numerical value – it’s been computed out to more than a million decimal places, but no matter how many we compute, there are always millions more. So there’s really no point in asking for the exact value of π. We can come as close as we want – or that we can afford (in terms of computer time).

Another way of looking at it is to realize that there is no pair of integers, a and b, such that a/b = π. There are a few that come close. The first is 22/7 = 3.142857…. (the decimal part repeats). Then comes 355/113 = 3.141592920353982….  (that one goes on for at least 30 decimal places with no repeat in sight)

This gives us a hint that there may be a way of calculating approximate areas and volumes that are “close enough”.

Start with the standard formula:

Circle area: A = πr2

It’s not always convenient to measure the radius directly. We can measure the diameter and divide by 2, but suppose we could work with the diameter directly:

A = πr2
r = d/2
A = π(d/2)2
A = πd2/4

which we rewrite as

A = (π/4)d2

We can look up the value for π/4; it’s almost exactly 0.7854.

So our area formula becomes

A ~= 0.7854d2

We use ~= to mean “approximately equal.

A very good approximation is

A ~= 0.8d2

this is within slightly less than 2%

It’s easy to multiply something by .8: double it three times, then divide by 10.

If d=12 inches, then A = 0.8*12 = 9.6 in^2.

Let’s see how that works out:

Suppose you have a garden with a 10-foot diameter circular area you want to plant with flowers.  You’ll need the area.   The actual area is 78.53982 square feet.  Using the “very good approximation”, we get 78.54 square feet.

The difference is 0.00018 square feet, or about 1/4 square inch.  And, it’s always a little lager than the exact value, so you never have to worry about running out.

We can also get the area if we know the circumference. The circumference of a circle is not so easy to measure – unless you’re working with a cylinder (like a tin can) and you need to get the volume. But the formula will be useful later on, so let’s develop it here.

A = πr2
c = πd
c = 2πr
r = c/2π
A = π(c2/4π2)
A = c2/4π

Moving right along to spheres

First, the formulas:

As = 4πr2

You may recognize the “πr2” part as the area of a circle with radius r. The surface area of a sphere is exactly 4 times the area of the circle whose center is  the center of the sphere (sometimes known as a great circle).  An old Greek guy figured that out, before anybody knew what π was.

Rewriting that to use the diameter:

r = d/2
As = 4π(d/2)2
As = 4πd2/4

Which simplifies to

As = πd2

That’s nice, but no help from the world of approximations.  But note that the area of a sphere is 4 times the area of a great circle.  We start over:

As = 4Ac

Now, we just go back to our circle area approximate formula:

Ac ~= 0.8d2

Plug that into the sphere area formula:

As ~= 4*0.8d2
As ~= 3.2d2

Finally, let’s find the area from the circumference. That’s a lot easier to measure, on a sphere. (Think of a basketball.)

As = 4Ac
As = 4(c2/4π)
As = c2

Calculating, 1/π = 0.3183, which we can round off to 0.3, or

As ~= .3c2

This gives a result that is just 6% too big.

Finally, the volume of a sphere:

Vs = (4/3) * πr3

From the diameter:

r = d/2
Vs = (4/3) * π(d/2)3
Vs = (4/3) * π(d3/8
Vs = (π/6) * d3

Calculate π/6 = 0.5326 Using 0.5 is too far off, so we’ll need to use 0.53:

Vs ~= 0.53d3

Now let’s step back and ask if it’s possible to calculate the area of a circle without using π.

Start with the basic formula

A = πr2

Rewrite that as

A = πr*r

From

r = d/2

replace one of the r’s:

A = π(d/2)*r
A = πd*r/2

But πd = c, so

A = c*r/2

The trick is that π is contained in c – because c = πr.

Ikea’s Magic Elevator

Seen in an Ikea store in Southern California:

 

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The Essence of Zen

Zen concepts can be difficult for Western minds to comprehend. In order to help us understand the essence of Zen, I’m providing this helpful explanation, which will save you thousands of hours of meditative exercises:

That that is, is not; that that is not, is.

The Glamorous Grammarian

Digging into word origins is fascinating, and can be rewarding, especially when something new and unexpected pops out. That was the case with these two words: grammar and glamour.

In modern usage, the two are unrelated. One refers to the rules for combining words of a language into correct sentences; the other, to an attractiveness (usually in women) that includes an alluring, elusive quality.

I leave it as an exercise for the reader to determine which is which.

Back in the 1930s through the 1950s or so, many Hollywood actresses were glamorous. That quality of glamour sustained many a soldier, sailor, and Marine during World War II. Betty Grable was one of the better-known glamorous actresses. Her pin-up photos graced many a ship or submarine’s wall, many a foxhole at the front, and many a makeshift aircraft hangar around the world.

I can’t think of any Hollywood stars of today with that quality. The elusive and mysterious aspects have largely disappeared.

Nowadays, even grammar seems to be becoming a lost art. I wonder how many high school graduates can name the eight parts of speech (or is it seven?…… (or that there is even such a thing)), or know the obscure notion of subject/verb agreement.

Considering how different in meaning these two words are, it came as a surprise that they stem from the same source.

Here’s the etymology for “grammar”, from an online Oxford English Dictionary:

late Middle English: from Old French gramaire,
via Latin from Greek grammatikē (tekhnē) ‘(art) of letters’,
from gramma, grammat- ‘letter of the alphabet, thing written’

And for “glamour”:

early 18th century (originally Scots in the sense ‘enchantment, magic’):
alteration of grammar. Although grammar itself was not used in this sense, the Latin word grammatica (from which it derives) was often used in the Middle Ages to mean ‘scholarship, learning’, including the occult practices popularly associated with learning

The common root of “grammar” and “glamour” is a word meaning “occult practices”, perhaps including witchcraft. Few witches in literature are described as glamorous, but a recent depiction of this one comes pretty close. And of course, Glinda the Good Witch

Another related word is “grimoire” (one rarely used, except perhaps by Harry Potter fans):

noun
a manual of magic or witchcraft used by witches and sorcerers.
Origin:
1850–60; < French, alteration of grammaire ‘grammar’ < Old French gramaire; see grammar

“Glamour” is one of the few words (perhaps the only one) that didn’t change from the “-our” lending to “-or” (as with “colour”/”color”) when it came to the States.

Typing today

Touch-typing has been replaced by thumb-typing:

 

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Easter Sunday calculation

Easter Sunday falls on a different day each year.  It can be as early as March 22 (the last one was in 1818; the next in 2285), or as late as April 25 (the last in 1943; the next in 2038).  This is because the date of Easter Sunday is tied to the lunar calendar.

UPDATE: Thanks to Bob Wright for spotting an error in one of the instructions. This is the corrected versiom.

The official definition is

‘Easter is the first Sunday
after the first ecclesiastical full moon
that occurs on or after March 21’.

Why March 21? That’s the first day of spring, or the Vernal Equinox. Here’s a detailed article on the date for Easter.

In 1800, the mathematician Gauss worked out the details for finding the date of Easter for any given year (after about 325). His algorithm accounts for lunar cycles and leap years,  among other things.

Since this is close to tax time – and may of you have probably been filling out the many and varied IRS forms – I thought it would make sense to present Gauss’ algorithm in a tax-form format.

Each line involves only one calculation, to reduce the chance of error.  I’ve included the results for 2013. (Easter is March 31).

If you print out the page, there’s space on the right to go through the calculations for any other year. (Start with 2013 to make sure you get the same results.)

You can check your results with tables of Easter dates.

1 Write the year 2013
2 Divide Line 1 by 19 and write the remainder [see Footnote] 18
3 Divide Line 1 by 4 and write the remainder 1
4 Divide Line 1 by 7 and write the remainder 4
5 write the first two digits of Line 1 20
6 Multiply Line 5 by 8 160
7 Add 13 to Line 6 173
8 Divide Line 7 by 25 and write the integer part 6
9 Divide Line 5 by 4 and write the integer part 5
10 Subtract Line 8 from 15 9
11 Add Line 10 to Line 5 29
12 Subtract Line 9 from Line 11 24
13 Divide Line 12 by 30 and write the remainder 24
14 Add 4 to Line 5 24
15 Subtract Line 9 from Line 14 19
16 Divide Line 15 by 7 and write the remainder 5
17 Multiply Line 2 by 19 342
18 Add Line 13 to Line 17 366
19 Divide Line 18 by 30 and write the remainder 6
20 Multiply Line 3 by 2 2
21 Multiply Line 4 by 4 16
22 Multiply Line 20 by 6 36
23 Add Line 20 to Line 21 18
24 Add line 22 to Line 23 54
25 Add Line 16 to Line 24 59
26 Divide Line 24 by 7 and write the remainder 3
27 Add 22 to Line 19 28
28 Add Line 26 to Line 27 31
29 If Line 28 is between 1 and 31, this is the date of Easter Sunday in March. Otherwise, continue to Line 30
30 Add Line 19 to Line 26
31 Subtract 9 from Line 30
32 This is the date of Easter Sunday in April

Footnote about remainders: If you’re doing all this with pencil and paper, you’ll get the remainder as a result of the division. But pencil and paper can be error-prone, so it’s probably better to use a calculator. Here’s how to get the remainder using a calculator:

Take Line 2, for example: “Divide Line 1 by 19 and write the remainder” Line 1 is the year, so we want the remainder of 2013 divided by 19. (For mathematicians and programmers, “mod(2013,9)”.)

Enter 2013.
2013
[divide by 19]
/
19
=
Subtract the integer part:

105
[multiply by the divisor]
*
19
=
See 18, the remainder.

Also, if the dividend (the number) is smaller than the divisor, the remainder is the number. For example, the remainder of 23 divided by 30 is 23.