Deriving Simpson's Rule

kw: mathematics, numerical integration, derivations

I've been teaching basic calculus to my son over the summer (can't believe it is almost over). We've had a look at the anti-derivative, AKA the indefinite Integral, and its use for summing a continuous function. Of course, if the function you want to integrate isn't a simple algebraic function, and you can't find it in a "Table of Integrals" anywhere, what do you do? Use numerical integration. So, we did a couple of sessions with the Trapezoid Rule, and how it can be accelerated by repeated use. Then, I told him we'd look at Simpson's rule, which has better accuracy and converges more quickly when accelerated.

I went looking on the web for a derivation, and found dozens of pages that say, "Now, for more efficient summations, we'll introduce Simpson's Rule. We present it without derivation..." or "...derivation of Simpson's Rule is beyond the scope..." So, I thought it over and decided, it can't be that hard...and it isn't! A simple picture illustrates the method. (Click for a larger image)

The figure shows how we analyze the total area in three sections:

• At the top, an area between the parabolic segment we are fitting through the three points (X0,Y0), (X1,Y1), and (X2,Y2) and the base of the parallelogram shown, with height Hp and width W = X2-X0.
• Just beneath, a Triangle with Altitude Ht and the same width.
• At the bottom, the Rectangle that fits between the Triangle and the X-axis, having the same width.

In Analytical Geometry (part of Pre-Calculus at most High Schools), we learn that the sagitta of any parabolic section (Hp in this diagram) of a certain width is the same, anywhere on the parabola. Further analysis shows that the area between the parabolic section and its enclosing parallelogram is constant. Thus the area between the parabolic section shown and the base of the parallelogram—the sloping line through the midpoint (X1,Y1)—is W·Hp/3.

Now we determine the heights of the three sections, Hp, Ht, and Hr:

• Hp = ½(Y0+Y2) - Y1 = ½Y0 - Y1 + ½Y2
• Ht = Y2 - Y0
• Hr = Y0 - Hp = ½Y0 + Y1 - ½Y2

This allows us to find the three areas:

• Ap = Hp·W/3 = W·(Y0/6 - Y1/3 + Y2/6)
• At = Ht·W/2 = W·(Y2/2 - Y0/2)
• Ar = Hr·W = W·(Y0/2 + Y1 - Y2/2)

Adding these, it is best to use W/6 for the common prefix, so

• A = W·[(Y0 - 2Y1 + Y2) + (3Y2 - 3Y0) + (3Y0 + 6Y1 - 3Y2)]/6

From which we find:

• A = W·(Y0 + 4Y1 + Y2)/6

And this is Simpson's Rule.

A Moral Voice in Fiction

kw: book reviews, fantasy

Many consider Orson Scott Card a writer of Science Fiction. Most of his early writing, up to the mid 1990s, is in SF settings. Yet the stories he tells could be retold in any period of history, on Earth as well as any imaginary planet. Many would lend themselves as well to a purely fantasy setting.

No matter what setting Card uses, his theme is ethics. He explores the limits of goodness, by itself and in conflict with evil. Prior to reading Magic Street, his latest book of contemporary fantasy, I'd read all the Ender series and Homecoming series books. The former series I consider a saga of atonement, the latter one of sacrifice, particularly the limits of altruism in the face of genuine evil. And, in Homecoming's denouement I was quite relieved when the pathologically altruistic protagonist yielded leadership to someone more pragmatic.

Magic Street is, to me, about redemption through transformation. Strip away the magic and mystery, and we find a protagonist who must risk his very self, having no guarantee of success. If I go into more detail, I'll spoil it, so read for yourself.

This theme is the core story of the Christian life portrayed in Paul's epistles. Though a believer's eternal salvation is assured through repentance (metanoia in Greek, the change of our thinking), the salvation of the soul proceeds in time through transformation (metamorphosis in Greek, the change of our psychic constitution). This determines the level of reward a saved person will gain upon being brought into glory through redemption of the body (metasoma in Greek, bodily transfiguration). As Paul warns, "star differs from star in glory."

The author is well read in both Christian and Mormon scriptures, and is an active Mormon. More than any other writer, I find him most skilled at presenting moral progress as his overt theme, without excessive preachiness nor the over-sentimentality that characterizes "religious fiction." For Mr. Card, religion is not mechanical, and his writing shows the fallacy of mechanicalism (AKA fundamentalism). Faith must be living, and like the book of Esther, though his writings don't mention God, God is present everywhere. I like that.

What is a Moderate, Really?

kw: book reviews, politics

I really hate politics. I find that it brings out the worst in almost everybody. It takes a really great leader to rise above politics, and none has done so consistenly. The two who come to mind, who came the closest to doing right consistently, regardless of political influences or ideology, were Churchill from 1938 onwards, and Lincoln from 1860.

I sometimes say, "Politics is forcing people to do what most of them would do anyway." Politics is about power, and make no mistake, power corrupts, always, everybody. I think of myself as a good man (but who doesn't?), yet when I was given power I abused it. I had to give it back to prevent further abuse. I have never known anyone to use power without abusing it. The best among us abuse it only a little, but be clear, nobody has wielded power free of abuse.

These thoughts were triggered by reading Christine Todd Whitman's new book, It's My Party Too: the battle for the heart of the GOP and the future of America. I read the large print edition by Thorndike Press. Mrs. Whitman, past Governor of New Jersey and most recently past Director of the EPA, is a moderate Republican...and I can't avoid the quibble, "whatever that means."

I am a Republican. That's how I usually vote. Now, I must ask myself, what kind of Republican am I? Am I conservative, neo-con, moderate, liberal, Lincolian, Reaganesque...?

Early in the book, the author lists a few items that define the moderate Republican stance:

• Financial reponsibility and prudence
• States-rights advocate with respect to "choice" and the definition of marriage
• Environmentally responsible (what Teddy Roosevelt would have called Conservationist)
• Cooperative rather than belligerent foreign policy

Sounds good. Even better, she quotes Arnold Schwarzenegger's 2004 speech: "If you believe that government should be accountable to the people, not the people to the government...if you believe a person should be treated as an individual, not as a member of an interest group...if you believe your family knows how to spend your money better than the government does...then you are a Republican."

She also quotes Jefferson, "That government governs best that governs least." And, defending her pro-choice and pro-gay-marriage stance, she quotes Goldwater's qualification of Jefferson, "...and stays out of the impossible task of legislating morality." Stay tuned for a separate rant on this stupid statement.

I personally prefer a very conservative foreign policy coupled with a somewhat liberal domestic policy. I believe the best defense is a strong offense, so I am in favor of the wars we have carried out in Afghanistan and Iraq. If we must, I am in favor of war with Iran and North Korea. I am in favor of defending the sovereignty Taiwan even if it means war with China. On the other hand, I don't believe we will succeed in the state-building side of our Iraq policy. We can't nursemaid the world. So superpower is super enough to do that.

Domestically, I believe first in personal responsibility, based on the Biblical words, "Whoever is not willing to work, do not let him eat," meaning don't feed a deadbeat. But that is not all the Bible says; Jesus said, "The poor you always have with you," meaning there are some whom we must care for. Therefore, while I am in favor of caring for our poorest, I believe every social welfare program must be coupled with education and training, as a requirement for receiving benefits.

Another of my proverbs, quoting Mark Twain: "Someone who is unwilling to learn has no advantage over someone who is unable to learn." Those unwilling or unable to learn belong in an institution where they are totally controlled. Those who find this an "attitude adjusting" adventure and decide maybe learning isn't so bad, should have every opportunity to do so, to become independent citizens.

I came to the book prepared to deplore it. I came away very much impressed. Mrs. Whitman has it exactly right, that both parties have been taken over by radicals. The political process has become so polarized that disagreement has been replaced by demonization. Well, that happens about every second generation. You oughta read some of the stuff people wrote about Lincoln in 1862-4...and some of the stuff he and others wrote back! And don't even mention how the British Parliament carries on, particularly in Commons.

For some reason, we in America tend to cycle between collegial debate and mad-pitbull slathering. I think the Brits do it better. Of course they have a half a millenium head start on us, learning how to have energetic debate, ranging from the occasional flash of logic to ad hominem and other fallacious attacks, rambling around a generally warm level of discourse, all the while getting the business of government done.

Oh, well. What does our own history show us? Three recent Presidential elections were overshadowed with the public's disgust:

• Gerald Ford lost to Jimmy Carter mainly because Richard Nixon had so besmirched the GOP.
• Carter lost to Ronald Reagen for first defining the "misery index" then showing us how to drive it sky-high.
• George HW Bush lost to Bill Clinton mainly because he was perceived as a betrayer: "...read my lips..."

But the others in my memory showed that the public tends to prefer the more moderate candidate. My main political philosophy is that people prefer to be left alone, and aren't long enamored of anything that may mean they need to work harder. Note to politicos everywhere: Don't stir my passions, just show me how I can live better and raise more fortunate children. About 70% of the populace will vote for whoever they percieve is least extreme.

Wise candidates know they must run in the primary as rather liberal Democrats or rather conservative Republicans, but in the general election as more moderate. Some today consider Ronald Reagan as a moderate. That is just because he is being compared to Sean Hannity. In 1979-80 Reagan was generally perceived as a flaming conservative, but there was no way most of us were going to let Carter have another whack at us. In 1959, Jack Kennedy was viewed with much alarm, but Dick Nixon was seen as even worse. Today Kennedy is seen as a conservative! In spite of the fact that his surviving brother is the leading Socialist in Congress—fact is, Teddy K. is more socialistic than any of the avowed Socialist Party members who have run for Congress.

The upshot is, I very much sympathize with Mrs. Whitman. But I still happen to like George W Bush better than any of the "moderate" Republicans who might nurse Presidential hopes. Think about this: if Dubya were REALLY conservative (as currently defined, primarily by the Religious Right), we'd already be drilling in ANWR, Roe vs Wade would be on its way out, state legislatures would already be debating the merits of a "heterosexual-only marriage" amendment, Biology teachers everywhere would be jailed if they breathed the word "evolution," and the Department of Education would have been disbanded. Oh, yeah, and our airport security would resemble that at El Al.

Have no fear, Mrs. Whitman. By 2008 or 2012 the public will tire of the Religious Right and the GOP will shed their influence like a torn glove. And, most likely, the "hate Bush" crowd, finding itself without a target, will finish imploding and the Democrat party will similarly shed the neo-Clintonians. You, Mrs. Whitman, seem to know how to be forceful without stridency. I hope you run for President.

My Heaven could be Your Hell

kw: utopias, dystopias

I happened to read Brave New World by Aldous Huxley, because my son had to read it to prepare for Junior English. Allow me to grump a bit, that this book really ought to be rated PG-21. I don't think it is suitable input to the formative teenage years, and not just for its sexual content. That is such a blatant caricature of the "free love" advocated by some 1920s Suffagettes, most teens can see through it. But some can't, and they'll be influenced; positive influence being unlikely.

I think I might be getting mature, as I approach 60, so I could tell that everything is a caricature. Huxley was presenting an admittedly false dichotomy (as his 1946 foreword notes) between the statist control and libertarianism. His "Savage" concludes in despair and suicide, a conclusion foregone in the existentialist tenor of the times, and explored further by Camus. Huxley did not succumb to the despair of existentialism, but surmounted it in a quest for spirituality and transcendence in his later writing.

As great writing does, this book can help us learn ourselves better. It also reveals the writer, as he uses the writing to work through difficult aspects of his own relationship with the world and where it might be going.

It is largely due to writers such as Huxley, Camus, Orwell, and other trend-extrapolators that society is able to avoid following risky trends to their logical extreme. Even more, looking back over history, every trend has buried within it a counter-trend; things tend to gravitate back to a middle-of-the-road situation over time.

Whatever the mighty of the world may want for us, most of us just want to be left alone.

Nancy Drew Grows Up

kw: book reviews, mysteries, parodies

I grew up reading a great many sorts of books, but I particularly remember the Tom Swift series (original, not "Jr") and the Hardy Boys. I didn't pay much attention to Nancy Drew or the older Perils of Pauline; they were "girl stuff." Forty years later, my son read at least a hundred of the newer Hardy Boys series. We used to joke that, with each book covering a few day's time, the Hardys must have had a time machine to fit all that into the year they were 17 and 18. Later he read (probably) everything of Encyclopedia Brown.

Confessions of a Teen Sleuth by Chelsea Cain brings us up to date on the life of Nancy Drew, her college years—when she roomed with Carolyn Keene, her own Watson—marriage, later life and final adventure at age 82. We get a glimpse of the destiny of the Hardys, Tom Swift Jr., E. Brown and others.

The parody is designedly self-conscious, well enough written for that to add amusement rather than detract. There was a style to writing for the teen market of the 1920s to 1950s, and Ms Cain has captured it well. A fun read.

Peregrinations

kw: book reviews, criminal activity, ornithology

On the Wing: To the Edge of the Earth with the Peregrine Falcon by Alan Tennant is a picaresque narrative of impersonations, theft, and various other examples of laws bent and sidestepped—and broken—in the service of being the first and only team to actually follow the migration of a few radio-tagged peregrine falcons.

All the action happened in the mid- or late 1980s, on the heels of an Army-sponsored project to radio tag falcons at So. Padre Island, TX, and follow the first few miles of migration using aircraft, to get a first guess as to their migration route. A satellite tracking system was in the works, but would not fly for a few more years. Tennant persuaded a pilot, who clearly hankered to follow 'a bit further, at least,' to attempt to follow a tagged falcon all the way to her Arctic nesting place. After succeeding in this, the following year they follow another into Mexico, Belize, and Guatemala. They don't quite finish the route, but learn enough to know the terminus.

The book contains much meditation and a little science. I must admit, the science is good. But I find the method rather more intrusive than I'm comfortable with. I greatly applaud the satellite tracking system that is now in use, now that the radio tags have been developed to be almost wholly innocuous.

While there were surprises in Tennant's discoveries, it really isn't work that needed doing, and has been much superseded. This is a story of derring-do with a scientific veneer.

How Science is Done

kw: opinion, observations, scientific method

While it is on my mind. Some of the nonfiction books I've read this past year contain side narratives on the process of scientific process. They touch a deep chord with me, because of experiences I'll cover only in part.

For my dissertation research, I studied heat flow in geologic settings. At the same time, I had two consulting projects, one for a mining company, one for a hydrologist. At depths exceeding 7,500 feet in one mine, I was surprised to find huge amounts of water flowing about. I recalled other mines I'd visited (there have been many), and water was always in motion. Then, for the hydrologist, I participated in a survey of groundwater in all formations to a depth of about a kilometer under an entire western state. I remember being told that the "impervious" layers between the aquifers were really just "less pervious". The analogy was, an aquifer is like a steep slope down which water can freely flow, while a "less pervious" layer, an aquitard, is like a much gentler slope, from which water will flow aside if it has a steeper slope to follow.

Putting these together with my research, I soon determined that most heat flows in the Earth's crust, not by conduction—like heat moving through a metal—but by the flow of heated fluids, mainly salty water, between hotter and cooler regions.

Another factor: at the time (70s and 80s) there was much debate over the best way to store spent nuclear waste. A company with much influence in the upper Midwest was making a ton of money doing studies of various "dry" formations, such as salt and granite, to see whether they could safely store 'used' uranium and plutonium for a few million years.

Now, I had concluded that the ratio of convected to conducted heat (Peclet's Ratio) was nearly always greater than one. A colleague and I wrote a paper showing that, in the least porous rock known, ordinary hydraulic pressures will move water at rates of a meter or so per century, and this will carry more heat around than the "normal" thermal conduction through the rock. Guess what happened when we tried to publish? Guess what happened to those chapters of my dissertation?

I discovered the human side of science. It is like the elephant in the corner. It is much bigger than the 'objective' side.

Over the next twenty years, first one, then another, part of my work was duplicated by others—who dwelt in less hostile climes, scientifically—until the whole is now 'common knowledge' among hard-rock geologists.

Perhaps you know the Baconian model: make lots of observations, organize them, and deduce trends you can test. Based on more observations, guided by the trends (models or hypotheses), refine until you can formulate a theory. Do you know how most scientific progress is made? By someone who propounds a theory that irritates someone big. The key is to get it into print. If nobody responds, keep working and print more. Eventually, someone will respond, and the best possible result is if that someone attacks the new theory. If you can get a media battle going, so much the better. Once you have enough in print, publish a "synthesis" monograph. At that point, there will be a significant body of irritated, even enraged scientists out there, who will bust their asses trying to prove the theory wrong. Maybe they will, and maybe they won't disprove it, but lots of good science will be done.

Just by the way, the most recent issue I've seen of "New Scientist" contains an article touting the work of a few scientists who find much is lacking in the Standard Cosmological Mode, which includes mainly the Big Bang, Inflation, Dark Matter, and Dark Energy. They think it is all bunk, and they've managed to get into print. I gleefully await the outcome.

Our Favorite Neighbor

kw: book reviews, spirituality, faith

I was 20 when Mister Rogers' Neighborhood began to air in 1968. I'd been raised on Howdy Doody Time, Mickey Mouse Club, and Captain Kangaroo. This new show seemed much too tame. Many years later, when my son was two, I began to watch two shows with him: Mister Rogers' Neighborhood and Sesame Street. About the last, I was on occasion uncomfortable with the content, but never about Neighborhood.

By the time he was seven or eight, he'd outgrown the show. Not long after that, I heard an on-air interview with Fred Rogers, and found he was an ordained minister, and that the show was his ministry. He carried it on nearly to the end. Had he been less frail, I think he'd have been pleased to die in the saddle. As it was, he experienced about a year of retirement before his death in 2003.

Long-time Rogers family friend Amy Hollingsworth has gathered her own spiritual and practical impressions of Fred Rogers for us, in her book The Simple Faith of Mister Rogers: Spiritual Insights from the World's Most Beloved Neighbor. The publisher is Integrity Publishers.

A train runs on two tracks. The author has organized her material around the trinity of heart, eyes, and hands. However, I was most struck by two spiritual influences—among many in his life—that seemed to guide Fred Rogers. Firstly, the proverb of St. Francis: "Preach the gospel at all times. When necessary, use words." Secondly, Dr. Wm. Orr, Fred's chosen mentor in Seminary, who showed that everyone is either an advocate or an accuser: it is our choice which we will be, in every situation. On these two tracks Fred Rogers guided his life.

One cannot read this book entirely dry-eyed. There is much that is touching, and more than touching. The revelation of a truly spiritual man draws an exquisitely agonizing call from the heart, to grow to such stature, to find pleasure in pleasing God. There is a tremendous difference between religion and spirituality. Mr. Rogers learned that well, and taught that...sometimes with words.

I find much of modern religion to be syrupy sentimentality. There is much sentiment in Amy Hollingsworth's book, but none of it is syrupy, and all is fitting.

The Political Gap in the War on Terror

kw: opinion, terrorism, political correctness, bill of rights

As I write this, reports of more "incidents" on buses in London are in the news. Strangely, newsmakers everywhere minimize any report that the bombers of two weeks ago were of "Mideastern" ethnicity or followed the "Islamic" religion. Even less is being said today.

When I was in Jr. Hi, a minor gang got started in my school. It was just a clique run out of control at first. But it got ugly pretty quickly. One of the teachers was wise. He knew who the leader was, so he went to see him, got him alone, gut-punched him, slapped him up against the wall, and said something like this: "This is a sample. Every time one of 'your' fellows gets out of hand, I'll do something worse to you. Don't tell me they are not in your control. That's not my problem, it is yours. I am not alone. I have friends, too." He only had to act one more time.

This is, in effect, the message sent to other terrorist leaders by the deposition and capture of Saddam Hussein. It will need to be sent a few more times. There are 'radical islamic' leaders who daily preach sedition against Western governments, while living in the US, England, and other nations. Why do we not arrest them? Only in the name of policital correctness. We need to start with them, and work our way down. You don't kill a hydra by attacking the multiplying biting heads; you must go for the heart.

It has always been a crime, even under the Bill of Rights, to speak sedition or treason in the U.S. If we are to win the War on Terror, we will follow the laws of the land, and abandon political correctness.

Real Priests Must Marry

kw: opinion, priesthood, pedophilia scandal

The "pedophile priests" scandal continues to generate news. To those who know history, it is only the latest in sixteen centuries of sexual misconduct by priests. Throughout the medieval period, if any priestly conclave was planned in a town, the townspeople sent away their daughters and younger sons to stay with relatives for the duration. This shame of Catholic dogma has been an open secret through its entire history.

I won't rehearse the history of the celibacy requirement. Rather, I simply show that it is wrong. Those who God has called to celibacy are very, very few. If a man or woman does not have a gift of singleness, marriage is required to reduce the temptation to sin. But we need to see what the Apostle Paul wrote. Recall that a Catholic priest is considered an elder, and if elevated, a bishop 0r overseer (We'll set aside for today the fact that in the Bible, every elder is a bishop, and every bishop is an elder: one is the person the other is the function).

Paul wrote twice about the qualifications of an elder or overseer:

I Timothy 3:1-7
1 Here is a trustworthy saying: If anyone sets his heart on being an overseer, he desires a noble task.
2 Now the overseer must be above reproach, the husband of but one wife, temperate, self-controlled, respectable, hospitable, able to teach,
3 not given to drunkenness, not violent but gentle, not quarrelsome, not a lover of money.
4 He must manage his own family well and see that his children obey him with proper respect.
5 (If anyone does not know how to manage his own family, how can he take care of God's church?)
6 He must not be a recent convert, or he may become conceited and fall under the same judgment as the devil.
7 He must also have a good reputation with outsiders, so that he will not fall into disgrace and into the devil's trap.

Titus 1:5-7
5 The reason I left you in Crete was that you might straighten out what was left unfinished and appoint elders in every town, as I directed you.
6 An elder must be blameless, the husband of but one wife, a man whose children believe and are not open to the charge of being wild and disobedient.
7 Since an overseer is entrusted with God's work, he must be blameless—not overbearing, not quick-tempered, not given to drunkenness, not violent, not pursuing dishonest gain.

From these two passages it is clear that an elder or overseer must be a married man with children, and he must demonstrate through his success in raising his children that he is fit to manage the church.

Over against these two passages, can be laid thousands of pages of the writings of the "church fathers," regarding celibacy. However, none of them is inspired by God; Paul's writing is. To a Bible believer, not a single word of "tradition" is authoritative. And the Bible is clear. Whether those who lead the church are called priests, elders, bishops, overseers, parsons, pastors...regardless, they are required to be married men who have successfully raised children.

Birdsong at Audubon

kw: book reviews, birds, bird song, birdsong, natural history, naturalists

Is there anything more fresh than walking through the cool morning air at, or just before, sunrise, hearing the morning chorus of birdsong? I confess it has been too many years since I have done so. This evening, though, taking a walk near dusk, I was treated to an evening chorus no less sweet or varied. There is a patch of woods near home, between one neighborhood and the next, as is the fashion in Eastern suburbs. It has been pretty well left alone since this development was built up some fifty years ago. This patch partly abuts a schoolyard, so we have our choice of a grassy sward or a running track to walk on.

I've noticed the birds and their singing in the past, but tonight I had different ears, all from reading Birdsong: A Natural History, by Don Stap. Perhaps half of the book is composed of information about singing birds and some history of the study of songbirds and birdsong. The structure of the book, though, is a picaresque history of the career of Don Kroodsma. The author has been privileged to accompany Dr. Kroodsma on a number of expeditions and excursions to record birdsong and study singing birds.

The recording itself is one thing. Making something useful of it is quite another. For that, one uses a sonograph, or sonic spectograph. Here is an example of the resulting sonogram.

The vertical dimension is frequency, with higher notes at the top. The scale ranges from 2,000 to 9,000 Hertz (1 Hertz = 1 cycle per second), for few bird sounds are outside this range. 9,000 Hz is an octave or two higher than most people can whistle. That's why it is hard to reproduce a bird's song. This song begins with a high chirp, a short beat of silence, then a lower whistle, followed by four chirps in the medium frequency range. To hear the sound and see & hear a few other examples, check out this Audubon Society site, which promotes Don Kroodsma's recent book The Singing Life of Birds. I've bought the book and will review it soon.

The song shown is called "Song A" because there are several dialects of the Eastern Towhee. Much work over the past twenty to thirty years has been done to record birds that have, and those that don't have, dialects, and to attempt to understand the differences.

Dr. Stap's writing recalled to me my early love of hearing birds sing. As a preteen with a new tape recorder, one of the first things I did was put the microphone outside my second-storey bedroom window and record a half hour or so of the birds waking up at daybreak. Now, 42 years later, if I can ever find that tape, it will be a valuable record. Birdsong begins with a description of the Macaulay Library of Natural Sounds. Many, many of the sounds record birdsong, including many dialects and variations.

It takes a great many recordings to produce a comprehensive collection of birdsong. Not only are there many singing birds—nearly all the passerines (perching birds), more than 5,500 species out of 10,000 or more total bird species—but some birds sing numerous different songs. Most (maybe, maybe less than most) have two calls, a territorial song and an alarm call. Many have a few or a few dozen songs, and there are some birds that sing hundreds or thousands of different songs. Some, such as the chipping sparrow, occasionally engage in repertoire contests, with a number of tiny birds, "dueling banjos" style, singing song after song to one another, very rapidly, often in a fixed order, in which the duelers either match one another note for note, or each jumps ahead to the next song, then the next...on through as many as four thousand songs.

Rather than grow this post to an hour's reading, I'll leave the rest for those who are interested, to check out or purchase Don Stap's quite enjoyable book.

Old Fiddles

kw: book reviews, aging gracefully

Click on this image to see a larger version.

The proverb is already old: "The older the fiddle, the better the tune." The violin pictured is an 1840 French 'court violin.' A princeling who found himself desiring a court orchestra would hire a concertmaster and order a batch of these: 20, 30, maybe even 50.

The ones made before 1880 or so have a carved head for a scroll. It is supposed to be the image of Casper Diuffelprogcar, who "standardized" the shape of the violin in about 1550. The size wasn't settled until later. The body of this one is about a half inch longer than a modern violin's. The sides and back are decorated with inlays of cities and musical instruments.

This fiddle has been in the family for six generations, beginning when a family friend gave it to my ancestor in 1850. The old gent had been to a concert in St. Louis (probably a variety show of the kind that became Vaudeville a generation later) and stopped one of the musicians to ask if he could buy the instrument. They settled on $100. Quite a princely sum for 1840 (Remeber in "Gone with the Wind," that the taxes on Tara were raised to an unheard-of $300?).

I've heard my great uncle play this violin. Its tone is very lively and sweet. It was "repaired" once by his uncle, who didn't get the neck back on quite right. It is coming loose, so until I get it restored, it isn't playable. But I am mighty gratified to have it. It has outlasted five generations of McKowns and Nyes, and with a bit of repair, could outlast five or ten more.

Would that we could be repaired and restored after decades of aging and wear. In this life at least, we can't. But we can choose to grow old either badly or well. It is up to us. "Once the cards are dealt, what matters is how we play them" (my own proverb).

Well, the original Ronald McDonald, Willard Scott, has collected notes and essays by a hundred or so friends and acquaintances, from Dr. C. Everett Koop to Pete Seeger; Mort Walker (Beetle Bailey) to Fred "Mr Rogers" to Jayne Meadows; and others well known to unknown. The title? You guessed it, The Older the Fiddle, the Better the Tune.

Don't expect anything in the way of organization, index, table of contents here. This is inspiriation, not education. Examples I like:

"...people have stopped trying to sell you life insurance." (John Updike at 70)

"...the older you get, the easier it is to shoot your age in golf." (Ralph Young at 80)

"My arteries are hard enough so that I don't have migraine headaches any more. " (C. Everett Koop at 84)

"The secret of long life is not perfection, rather the ability to adapt." (Jessie L. Dunn at 92)

"I took up painting with pastels at age seventy and thoroughly enjoyed it." (Edith H. Aschenbach at 100)

A fun read, and a bit of a leg up when you're feeling a bit over the hill.

Almost Natural

kw: book reviews, nature, simple living

I spent a summer in wilderness areas more than thirty years ago. It was the traditional Geology seniors' "summer field camp." About a month in eastern Nevada, and a month in the Sierra Nevada in California. In both places, the setting was beautiful, though the first was quite arid. I pitched my tent rather far from the others, both places, and pretty much took care of myself. We had a few common activities, meetings to correlate the edges of our maps, group hikes to places of common interest. But mostly I hiked about, planning to cover my assigned area about twice in the time I had.

I particularly liked the many pools, ponds, and lakes in the Sierra camp. I planned each day's loop to put me at a lake midday, where I'd take my daily bath (I use a biodegradable surfactant to bathe, not soap, to this day) and a swim, then lie about to dry off. I'd surface map on the way out, and collect specimens on the way back to camp. I walked 500 miles that summer.

I distinctly remember, near the end of the month in the Sierras, meeting a small group of backpackers who were hiking into the back part of the wilderness area I was just leaving (The Geology camp was just outside the boundary of the designated wilderness). I asked how long they planned to stay.

"A week!" one replied excitedly. "How long have you been here?"

"Almost a month." / "Wow! That's neat!"

I hesitated, then said, "No, it's not. Not really."

Nonetheless, after I left, I entertained fantasies of returning, maybe on a more permanent basis. I never have. In graduate school, I picked a campus that was near the Black Hills, where most Geology Departments across the West and Midwest have their field camps. I figured, if I lived there, I could get my field work done without spending any more time in a tent. I've done my share of living simply, catching or shooting most of my own food, and buying little. I never did try roughing it through a Sierra winter, not even a Nevada winter.

So, reading David Petersen's book On the Wild Edge: In Search of a Natural Life was a bit nostalgic for me. He has done for upwards of 25 years what I only dreamed about. And as I read, I said to him in spirit, "David, better thee than me." He has a chosen path, and is fulfilled in it. I have mine, and am fulfilled.

I have to say this, he is a compelling writer. He tells a story that you find in your dreams that evening.

Half the book is rumination, anecdotes, and deft descriptions. Half is polemic, in the good sense of the word. His edge is shifting, and he is calling the alarm, rightly so.

What is the edge? Consider two extremes, two men. One is the urban denizen of Yokohama or Manila who never sees meat that isn't shrink-wrapped, takes the subway, bus, or taxi everywhere, works hard, plays hard—usually in smoky clubs or noisy concert venues—, and hasn't seen a living tree since he was four. The other is a Japanese soldier who hid for fifty years in the Philippine jungle, living a life more self-reliant and remote than any Crusoe. Both are products of a technological culture, so we're not talking Third World subsistence here in the case of the hiding soldier.

Now imagine a spectrum, a gradation between these two extremes. The inner city grades to 'uptown,' to suburbs, to exurbs, semirural, rural, remote, and maybe a step or two more to reach 'untamed' jungle. Anywhere along this continuum could be The Edge, it depends on your viewpoint. For me, The Edge would be a step beyond suburban living. That summer in the tent was beyond it, for me, and I really don't care to return for more than a short visit.

My graduate school years, I lived right at my Edge. A pocket development, just outside the city limits of Rapid City, SD: eight houses on two to eight acres each, along one side of a country blacktop road. Across that road was a cattle ranch. Usually, we were upwind. My next door neighbor was also a rancher. His ranch was thirty miles further out. Each rancher owned and leased about 15,000 acres, and ran ten cattle to the square mile: 64 acres each. That was a bit more than the 50 acres that would barely support a cow. At various times in our yard, we had runaway calves (with horsemen chasing), pronghorn antelope, pheasant, plenty of songbirds, a few kinds of snakes, and—we found out after getting a cat—quite a variety of mice, voles, shrews, and other mini-mammals.

We planted a shelter belt of trees and bushes obtained free from the County Extension (4H): Russian Olive, Serviceberry, Plum, and Chokecherry. About 50 each. Each fall we went into the hills to pick berries, mainly chokecherry, to make jelly or jam. From a ridge on the ranch across the way I could see Mount Rushmore with binoculars. We got to know everyone within a mile's walk. That came to ten families.

I could drive twenty minutes and be in the midst of pine woods, on rocky ledges and forested slopes where I would like not see another person all day. I'd do what rock collecting or other measurements I needed for my research, whistling or silent, with only animal and wind sounds about. Ten minutes in another direction, and I'd be at Rushmore Mall. Near my office at the School of Mines, I could take a swim in an Olympic size pool.

For me, that is The Edge. I am glad there are people like David Petersen. His Edge is not too close to being a naked man in a cave. He lives with his wife in a cabin he built. They have a few modern amenities, and he makes what money they need writing on a laptop computer. He kills an elk per year, which takes care of their meat needs. Of course, that implies he has enough electricity to run a deep freezer that'll hold 300 pounds of meat. He had an ethic we'd all do well to consider, though few of us could emulate the resultant life.

To people like me, he's the canary in the coal mine. He is getting crowded out, on his mountainside near Durango. That's not good. If his new neighbors had any idea of respect for wilderness and wild things, maybe it wouldn't be too bad. But nearly all are city folk who move to the country, then bring the city, and city attitudes, with them. You can't be raised in the city and develop the kind of reflexes that are appropriate to rural or near-natural living. A pity.

Measuring Rod Hierarchy

kw: parallax, cepheids, cosmology

Reading of Henrietta Leavitt and her work with Cepheid variables nearly a century ago, got me thinking of stellar and galactic distance measurement. The "hierarchy" of measurement methods is really a set of parallel ladders, at least in places.

1. The scale of the Solar system.

The earth-moon distance, being about thirty times the Earth's diameter (or about 10x the circumference), was measured centuries ago. Measure the moons' position in early evening, then just prior to sunrise for a few successive days around full moon. This allows you to back out the effect of the moon's motion. Then you just need to know the size of Earth. Eratosthenes's measurement of 23,000 miles (this is approximate, because we don't know for sure how long his σταδιου was) for Earth's circumference thus yielded nearly 230,000 miles. The modern average (it varies, of course) is 240,000 miles or 385,000 km.

But, try as they might, nobody could determine a parallax for the sun until the telescope was put to use (though not discovered) by Galileo. It comes out to about 400 times the moon's distance, and today most of us learn 93 million miles, or 149 million km.

Much later, 28th Century, secular variation in the timing of the eclipses of Jupiter's moons coupled with early measurements of the speed of light yielded a parallel measurement of the distance to the sun. This distance, the Astronomical Unit, or AU, is our beginning baseline for measuring larger distances.

Within the Solar system, we often state distances in AUs: 5.2 for Jupiter, 19 for Uranus, and so forth.

2. Direct parallax of the nearer stars.

Alpha Centauri was not the first star to be 'parallaxed.' Stars known to have large proper motions (apparent sideways motion, year to year) were checked, and the first to be determined was found to be at a distance of about 9 light years. That is about 600,000 AUs. That means the long, skinny triangle they had to measure had an acute angle of 1/10,000th of a degree or about a third of a second of arc. Alpha Centauri, which came later, has a parallax of about 0.8 arc second. The smallest distance between two points that the human eye can see is about one arc minute, or 60 arc seconds. It takes a pretty good telescope to reliably measure angles smaller than an arc second.

The stirring of the atmosphere prevented accurate measurement of stellar distances beyond about 100 light years. In the past twenty years, orbiting telescopes, most notably the Hipparcos observatory, being above the atmosphere, have been able to provide accurate parallaxes within about 1000 light years.

So how to measure farther?

One idea was to measure star positions over many years, letting the motion of the Sun through the galaxy provide a longer baseline. This is called Secular parallax. Trouble is, you don't know the relative motions of the Sun and the stars you want to measure. Using a star cluster, such as the Pleiades or Hyades, you can use perspective to back out the needed relative motion. Such measurements have allowed the distance to a few clusters within 2,000 light years to be determined. This method is called statistical parallax.

This was enough to get the distance to a Cepheid or two...

3. The rôle of Cepheid variables

Henrietta Leavitt measured the light curves for hundreds of Cepheids in the Magellanic Clouds. Once she discovered the Period-Luminosity relationship, she had a 'law' of sufficient accuracy to determine that the Large and Small 'clouds' (LMC and SMC) were at different, but similar, distances. At the time, no Cepheid's parallax was known, so it wasn't known just how far away the Clouds were. Now they are known to be about 185,000 and 165,000 light years from here.

Shapley's first use of statistical parallax gleaned a couple of Cepheids, and they were found to be among the brightest stars in their clusters. This was good news. In fact, it is very good news, because Cepheids are bright enough to give us reliable information to a distance of about 50 million light years. So in my earlier post about Miss Leavitt's biography, the "joint" in the intergalactic yardstick provided by "her stars" fills the gap between modest intragalactic distances and rather great intergalactic, though intracluster, distances. A range of between 1,000 light years and distances 50,000 times as great.

4. Beyond the Local Group of galaxies

Going farther than that, we get into the "Realm of the Nebulae," to borrow the title of Edwin Hubble's book on galaxies (He never used the term 'galaxy' himself). He measured the spectrum of many galaxies, starting of course with the brighter ones. Before long, he realized something akin to Leavitt's realization: there was a something-versus-distance rule at work here. In her case, the period-luminosity relation uncovered a way to determine distance using Cepheids' brighnesses. In his, redshift and distance were correlated.

What is redshift? Oddly enough, it seems almost everything that can happen to starlight as it makes its way to an observer is one or another kind of reddening. Thin gas clouds scatter blue light away from the path between us and the source, leaving more green, yellow and red. Dust clouds absorb light pretty evenly across the spectrum, but also scatter blue and green more than yellow and red. So a star that you would classify as a very white type A based on its spectrum—which lines are present or absent—looks more like a type F or G based on its visibly yellow color, if its light has passed through some gas or dust.

Motion of a star toward or away from us causes a blueshift or redshift, but these don't change the visible color much. Rather, lines in the spectrum appear to be misplaced. For example, there is prominent red Hydrogen line in stars cooler than type F. If the star is moving toward us, the line will be shifted to a shorter wavelength; it is blueshifted. If the star is moving away, the line will be at a longer wavelength, redshifted.

Hubble found that the spectra of the galaxies were nearly all redshifted, and the farther away a galaxy was, the greater the shift. As mentioned, the overall color didn't seem much different; even though a speed of 10% the speed of light only shifts a yellow 580nm line to the orange-yellow 640nm, there is plenty of light that began around 530nm that is now at 580nm. Yet such a red shift, we now know, implies a distance of 1/10th the way to the edge of the visible universe, or more than a billion light years.

Since the time of Hubble, other distance measures have provided parallel checks on the redshift-distance relation. Type Ia supernovae, for example, have been studied in sufficient detail that we know they all have the same brightness (within a small measure of variation). The more things we discover, the better we are able to determine the distances of objects throughout the visible universe.

Beebe

kw: book reviews, naturalists, bathymetry

I had the good fortune not long ago to check out two books about William Beebe from the local library.

The first, The Remarkable Life of William Beebe: Explorer and Naturalist, by Carol Grant Gould chronicles Beebe's life. It is a good beginner's biography, and is the first biography one could call "authorized," because Ms Gould was persuaded to write it by Jocelyn Crane, Beebe's companion the last ten years of his life, and the holder of his journals and letters.

Beebe is of course best known to my generation for his book Half Mile Down, about the use of the Bathysphere to study creatures in midocean near Bermuda in the early 1930s. He was already one of the best-known scientists of his generation, and was famous for his writing and lectures by the time this picture was taken in 1917. In his early career, he was known primarily as an ornithologist.

A few years after the Bathysphere adventures, he was continuing oceanic studies, but moving more and more in the direction of focused tropical studies. He loved the tropics, and spent increasing amounts of time in tropical laboratories. By 1940 or so, his appearances in his office at the Bronx Zoo, shown in the second image, were becoming rather rare.

Beebe, usually called Will by all and sundry, was associated with the Bronx Zoo and the New York Zoological Society his entire professional life. He was occasionally sponsored by the National Geographic Society, particularly in the 1930s. He wrote articles for them and others, and monographs throughout his career, that are still classics in several fields.

Gould's book is comprehensive, very readable, and unblinking. We have her, and Miss Crane, to thank for presenting to us a fascinating study of a great generalist in a time of increasing specialization.

While his studies of birds, tropical ecologies, and netted sea life gained him great fame, what captured the imagination most—and still does—was the subject of Descent: The Heroic Discovery of the Abyss by Brad Matsen.

Beebe had been fascinated with undersea life since some youthful experiments with diving bells and helmets. In his fifties, frustrated with the difficulty of studying abyssal creatures brought up in nets and dredges—they were all too often pulped or exploded by pressure changes—he began to design a deep diving apparatus. His engineering skills were, in this regard, woefully inadequate.

Otis Barton was an engineer who knew his stuff. Scion of a well-to-do family, beneficiary of a generous trust fund, he was Harvard-educated in engineering and mathematics. He also had experimented with undersea apparatus, and planned to create a deep-diving submersible.

Beebe was his idol. When Barton read of Beebe's plans, he was at first depressed, thinking he would be scooped. Then he realized his design was likely to save Beebe's life. He gained audience with Beebe (through much trouble; Beebe had a crackpot a day come by with one scheme or another), and they became partners.

Matsen outlines in exciting detail the development of the initial, too-heavy Bathysphere, its redesign, and the studies Beebe and Barton carried out with it. For this section of Beebe's life, it greatly expands the story told more briefly by Gould. And it brings Barton to light, where he has been rather a cipher previously.

These two ambitious, complex, driven men, who each found the other hard to tolerate, managed to capture the world's attention with great skill and daring. I know, when I read Half Mile Down, more than forty years ago, Beebe's descriptions of the deep they observed were thrilling beyond measure. Beebe had a bit to say about Barton, but it took this book by Matsen to bring both men into a clear light, that I find a much more satisfying understanding of the risks they took and privations they suffered, to begin to open to us the depths of the sea.

Mr. Think Different, in his own words

kw: book reviews, scientists, letters, correspondence

I worked at Cal Tech in 1974 and 1975—as a machinist. I made parts for a new design of microwave telescope. One day, walking across campus, I caught sight of Dick Feynman striding across my path. I had no chance to greet him, and it was the only time I saw him. I wish I'd been bold enough to give chase and at least meet him.

I've been fascinated by Dr. Feynman (hereafter RPF, as he often signed his letters) most of my adult life. While in graduate school, I had a friend who owned a set of his Lectures on Physics. I read quite a number of the articles over a year or so. I recall that first one, though, on the Uncertainty Principle of Heisenberg. When I finished it, I had a visceral understanding of the principle. I have a set I bought, many years later, and this summer, my son is required to read through at least Volume 1.

My son was born late in 1988, just eight months after RPF died. That was nearly twenty years after I learned to use Feynman Diagrams to understand subatomic interactions. Now, sixteen years after he said, "I'd hate to die twice. It's so boring," his daughter Michelle Feynman has edited a wonderful collection of his letters: Perfectly Reasonable Deviations from the Beaten Track published by Basic Books. The material is arranged roughly in chronological order, one or a few years per chapter, and topically within chapters.

RPF is seen to be a master of both the long, witty, discursive essay-in-letter genre and the short, punchy note. Here is a piece of the former (To Dr. R. B. Leighton, my boss when I worked there):

"...There is one direction of computer science that has particular personal appeal to me. (It is called by the unfortunate name of 'artificial intelligence,' unfortunate because in the past several obviously naïve ideas went under that name.) Virtually all computer programs today simply follow step by step instructions—they do exactly what you tell them...The programmer does a great deal of work to write a program of instructions for a computer to do what you wish. To what degree can we use machines to help in this programmers work—ultimately to make machines which program themselves with little more information than we now give the progammer who at present makes the program?"

In succeeding paragraphs, he discusses language learning ("We are far from learning how it is done"), time-sharing computer systems and how they might compare with a person doing several things at once, and image processing; all in the context of discussing an offer to be made to a visiting professor.

My favorite among the latter form (to Dr. George Beadle, who'd offered him an honorary doctorate):

"Dear George,

Yours is the first honorary degree that I have been offered, and I thank you for considering me for such an honor.

However, I remember the work I did to get a real degree at Princeton and the guys on the same platform receiving honorary degrees without work—and I felt an "honorary degree" was a debasement of the idea of a "degree which confirms certain work has been accomplished." It is like giving an "honorary electricians license." I swore then that if by chance I was ever offered one I would not accept it.

Now at last (twenty-five years later) you have given me a chance to carry out my vow.

So thank you, but I do not wish to accept the honorary degree you offered."

I learned something I'd been only peripherally aware of. RPF gave much effort to evaluating textbook, particularly math texts. The title of the book is from his discussion—lament, really—of the "one way to do things" approach in nearly every text. Mathematics, in particular, must be done by any and all means, and every 'problem' has at least half a dozen methods of solution. Only one or a few may be considered, "elegant", but all will work. As a matter of fact, "checking your work" is really a way of saying, "OK, do this problem a different way to see if you get the same result," or ,"Now do it backwards, to see if you get back to the beginning."

Well, I may rant more on this later. I'd like to give one example though, of something nearly all of us learned one way, but can be done different ways: Multiplication.

Long Multiplication: This is the way we learned it. The only variation is the way carries are handled. Some above the middle section, some below, some digits, some more cryptic marks. Other than that the method is uniform. Yet it is not the only way to carry out multiplication.

Russian Multiplication may be apocryphal, but it is a method I remember seeing once or twice as a boy. Here is how it works:

Put the smaller number on the left, and leave room for a wide column on the right with the larger number.

Divide the smaller number by two repeatedly, discarding any remainder, until you reach one.

Double the larger number to match each entry on the left.

Cross out any line that has an even number in the left-hand column.

Then add the numbers in the right hand column, that are not crossed out.

The wonderful thing about this method is that it matches very closely the way computers add numbers. The doubling and halving correspond to the way computers keep numbers: in binary, so that 19 is 10011 and 23 is 10111. To double a binary number, shift it to the left and append a zero. To divide it by two, shift it to the right and discard any bits (1 or 0 digits) that fall off the end. The computer puts both numbers into Shift Registers, and shifts one to the left, the other to the right, adding the number in the first Shift Register to a third Sum Register every time a 1 bit drops off the second Shift Register. When all the shifting and adding is done, the result is in the Add Register.

Pascal's Multiplication method is a different way of arranging the digits, to carry out Long Multiplication. This works with numbers of any size, though it gets cumbersome with larger numbers; so does Long Multiplication, but it has the virtue of taking the smallest amount of paper. To use Pascal's method, first make a box, divided as shown, with diagonals from every corner, and leave plenty of room for a carry digit.

Multiply the digit above each box's column with the digit to the far right of its row, creating a 2-digit number that you split across the diagonal line as shown (2x1 = 02; 3x9 = 27. See how they are shown in the figure).

Once all the boxes are filled, add the numbers in each diagonal section, down and to the left. Put any carry digit near the border of the main box.

The digits at the bottom left of each diagonal section make up the result.

Why do I go into this discursion? To show that even in something we think of as a very standard math method, there are several ways of getting the same result. RPF made this point again and again in his reviews—complaints, really—of math textbooks.

I love the book.

The Joint in the Cosmic Ruler

kw: book reviews, stellar parallax, cepheids

The 2005 offering of the Great Discoveries Series of W.W. Norton & Co. is Miss Leavitt's Stars: The Untold Story of the Woman Who Discovered How to Measure the Universe, by George Johnson, a science reporter at the New York Times.

Miss Leavitt was a computer. From the late 1800s through the mid 1940s, a computer was a clerk, usually a woman, who computed primarily astronomical measures using pen and paper. Some observatories employed a dozen or more computers. There were perhaps six at Harvard in the early 1900s. Henrietta Swan Leavitt was the most prolific and exacting, and truly deserves the title Scientist in her own right. She it was who noticed the period-luminosity relationship for Cepheid variables in the Small Magellanic galaxy, and she who published it.

The book is thin, partly by design, but partly because there is so little to go on. We have the scientific data, of course, but very little personal information, and much of that concerns Miss Leavitt's illnesses. Born July 4, 1868; employed 1893 as a computer at Harvard—first as a volunteer; published "1,777 Variables in the Magellanic Clouds in 1908; died of cancer December 12, 1921.

At the time of her discovery, trigonometric parallaxes were known only for a few hundreds of stars, all closer than 50 light years. Over a generation, this was extended to 100, but other methods were needed to find the distance to the nearest Cepheids. Statistical parallax, developed by Shapley beginning in 1914, could provide good estimates within a thousand light years or so. It depended on accurate measurements of the proper motion of many stars in an open cluster.

Within Miss Leavitt's lifetime, no extragalactic measurements were made. It was in 1923 that the first Cepheid variable in the Andromeda galaxy was found by Hubble. Some decades were consumed with the discovery that old Population II variables were one-fourth the brightness of younger Population I variables. Once this was ironed out, the distance to Andromeda was determined to be about 2 million light years. Since then, Cepheid variables have established the distances of numerous within 40 million light years.

The book is not a biography, as I have said. It suffers from an insufficiently biographical approach. It is rather hard to pin down sequences, for the narrative is many-threaded, giving us capsule biographies of the many scientists who employed Henrietta Leavitt, or the results of her work. Nonetheless, it limns the career of a strong-minded, persistent, creative woman who filled in a crucial gap in the chain of measurement we use to determine distances to the edge of the universe.

Caving by Boat

kw: vacations, commercial caves, wildlife parks

We were away for a few days...unconnected, even! We went with a church group to central PA. One place visited was Penn's Cave. This shows the entrance from within the cave, on the boat—it is a water cave. You take the boat right through the cave, and come out the other end, shown next.

If we had walked the cave, it would have been a half hour walk, probably. The boating, the lake you spin 'round out the other end, and the traverse back to the beginning entrance, make it about a 50 minute trip. We enjoyed it very much.

Penn's cave has a nice variety of formations, and is a fully "living" cave. The formations are growing. On many of them you can see drops of water, depositing calcite as they evaporate. Be prepared to get a few drops on you! I was taught as a kid, "Stalactites are stuck 'tite' to the ceiling, and Stalagmites (note the g) are just a 'mite' weaker, so they stand on the floor." When the two grow together, they become a cave column. Penn's cave is young enough that it has few columns.

There are few things I find more beautiful than living cave formations. Some caves, such as Carlsbad and Mammoth, are stupendous in size, with lovely formations in some areas, though most are dead in them. Growing formations have an extra vibrancy.

After the cave trip, we went on the Wildlife Tour. The Penn's Cave ranch has a large area with native wildlife. The ones we saw were Bison, Longhorn cattle including those shown here, Deer, Elk, Gray Wolves, a very old Timber Wolf, and some bears. We all particularly liked the sign they have near the bears.

Bears, more than other wildlife, quickly form a bond with their keepers. The ones we saw—three black bears, one the "cinnamon" color phase—are kept well fed, and also enjoy the treats the keeper gives them, including clover flowers. A few kids tried the flowers and found them a little sweet. Better'n twigs, I guess, if you're a bear.

They also have an exhibit of stuffed African animals. They emphasized that the hunting was done fully licensed, for preparing the exhibit. I still don't like the idea. There are more than sufficient exhibits of African animals. Such "scientific" hunts are really for revenue, which the countries sorely need. I prefer ecotourism, because an animal that bring $5,000 as a dead specimen, can bring much more than that as a photographic subject, over and over again.

Well, that bit of unpleasantness aside, it was an enjoyable trip. Sunday morning we met with the church in State College, just a few miles away. Lovely ending to a 2-day outing.