Looking Up


Astronomer Clyde Tombaugh assembled his first telescope from spare parts on his family’s Kansas farm — the crankshaft of a 1910 Buick, a cream-separator base, and mechanical components from a straw spreader. He used this to make sketches of Jupiter and Mars that so impressed the astronomers at Lowell Observatory that they gave him a job there.

Years later, after he had made his name by discovering Pluto, the Smithsonian Institution asked if it could exhibit this early instrument. He told them he was still using it — he was making observations from his backyard near Las Cruces, N.M., until shortly before his death in 1997.

“Its mirror was hand-ground and tested in a storm cellar,” wrote Peter Manly in Unusual Telescopes in 1991. “It’s not the most elegant-looking optical instrument I’ve ever used, but it is one of the better planetary telescopes around. … Because of its role in the history of astronomy, I would classify this as one of the more important telescopes in the world.”


The following story is true. There was a little boy, and his father said, ‘Do try to be like other people. Don’t frown.’ And he tried and tried, but could not. So his father beat him with a strap; and then he was eaten up by lions.

Reader, if young, take warning by his sad life and death. For though it may be an honour to be different from other people, if Carlyle’s dictum about the 30 millions be still true, yet other people do not like it. So, if you are different, you had better hide it, and pretend to be solemn and wooden-headed. Until you make your fortune. For most wooden-headed people worship money; and, really, I do not see what else they can do. In particular, if you are going to write a book, remember the wooden-headed. So be rigorous; that will cover a multitude of sins. And do not frown.

— Oliver Heaviside, “Electromagnetic Theory,” in The Electrician, Feb. 23, 1900

(When asked the population of England, Thomas Carlyle had said, “Thirty million, mostly fools.”)

Court Order

From Chapter 12 of Ken Follett’s novel The Pillars of the Earth:

‘My stepfather, the builder, taught me how to perform certain operations in geometry: how to divide a line exactly in half, how to draw a right angle, and how to draw one square inside another so that the smaller is half the area of the larger.’

‘What is the purpose of such skills?’ Josef interrupted.

‘Those operations are essential in planning buildings,’ Jack replied pleasantly, pretending not to notice Josef’s tone. ‘Take a look at this courtyard. The area of the covered arcades around the edges is exactly the same as the open area in the middle. Most small courtyards are built like that, including the cloisters of monasteries. It’s because these proportions are most pleasing. If the middle is bigger, it looks like a marketplace, and if it’s smaller, it just looks as if there’s a hole in the roof. But to get it exactly right, the builder has to be able to draw the open part in the middle so that it’s precisely half the area of the whole thing.’

How is this done? Inscribe a diamond within a square and then rotate it 45 degrees:

court order

Langley’s Adventitious Angles

Image: Wikimedia Commons

Edward Mann Langley, founder of the Mathematical Gazette, posed this problem in its pages in 1922:

ABC is an isosceles triangle. B = C = 80 degrees. CF at 30 degrees to AC cuts AB in F. BE at 20 degrees to AB cuts AC in E. Prove angle BEF = 30 degrees.

(Langley’s description makes no mention of D; perhaps this is at the intersection of BE and CF.)

A number of solutions appeared. One, offered by J.W. Mercer in 1923, proposes drawing BG at 20 degrees to BC, cutting CA in G. Now angle GBF is 60 degrees, and angles BGC and BCG are both 80 degrees, so BC = BG. Also, angles BCF and BFC are both 50 degrees, so BF = BG and triangle BFG is equilateral. But angles GBE and BEG are both 40 degrees, so BG = GE = GF. And angle FGE is 40 degrees, so GEF is 70 degrees and BEF is 30 degrees.

Even Up

rolling die puzzle

Suppose we cover a chessboard with 32 dominoes so that each domino covers two squares. What is the likelihood that there will be an even number of dominoes in each of the two orientations (horizontal and vertical)?

In fact this will always be the case. Consider the 32 squares in the odd-numbered horizontal rows. Each horizontal domino on the board covers either two of these squares or none of them. And each vertical domino covers exactly one of these squares. So the horizontal dominoes cover an even number of these squares (call it n), and the number of squares remaining in this group (32 – n) must also be even. This latter number is also equal to the number of vertical dominoes, so both quantities are even.

(By Vyacheslav Proizvolov.)


The young specialist in English Lit … lectured me severely on the fact that in every century people have thought they understood the Universe at last, and in every century they were proved to be wrong. It follows that the one thing we can say about our modern ‘knowledge’ is that it is wrong.

… My answer to him was, ‘… when people thought the Earth was flat, they were wrong. When people thought the Earth was spherical they were wrong. But if you think that thinking the Earth is spherical is just as wrong as thinking the Earth is flat, then your view is wronger than both of them put together.’

— Isaac Asimov, The Relativity of Wrong, 1989

(J.R. Deller Jr. wrote, “Education is the process of telling smaller and smaller lies.”)



“It was hard for me to believe. I would look down and say, ‘This is the moon, this is the moon,’ and I would look up and say, ‘That’s the Earth, that’s the Earth,’ in my head. So it was science fiction to us even as we were doing it.” — Alan Bean, Apollo 12

The Cat Gap

The first “true cat,” Proailurus, or “Leman’s Dawn Cat,” appeared about 30 million years ago. But from 25 to 18.5 million years ago, strangely few catlike fossils are found in North America. Biologist Luke Hunter writes:

Following the appearance of the dawn cat, there is little in the fossil record for 10 million years to suggest that cats would prosper. In fact, although Proailurus persisted for at least 14 million years, there are so few felid fossils towards the end of the dawn cat’s reign that paleontologists refer to this as the ‘cat gap’. The turning point for cats came about with the appearance of a new genus of felids, Pseudaelurus.

The gap may be due to changes in climate and habitat, the rise of competing doglike species, an unsustainable “hypercarnivorous” dietary specialization, or some other factor. Modern cats descended from Pseudaelurus.

Kitchen Aid

Three male offspring, aged 9–14 years, of one of the authors were observed to experience visual problems profound enough to imply functional blindness. The visual deficit was evident on almost every occasion when any one of the children of this physician went to the refrigerator and opened the door. The acute visual problem encountered was noted to be part of a consistent behaviour pattern, wherein a few seconds after the fridge door was opened a cry would be heard from the affected child of ‘Mum, where’s the milk?’

— Andrew J. Macnab and Mary Bennett, “Refrigerator Blindness: Selective Loss of Visual Acuity in Association With a Common Foraging Behaviour,” Canadian Medical Association Journal, Dec. 6, 2005

Benedetti’s Puzzle

This is interesting: In 1585, Italian mathematician Giovanni Battista Benedetti devised a piece of music in which a precise application of the tuning mathematics causes the pitch to creep upward.

Avoiding this phenomenon requires an adjustment — a compromise to the dream of mathematically pure music.