What moves in the sky, how do we know, and which model accounts for it best?
Level 1
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7 ideas. Read the hook, pick an answer, see the picture.
The Sky as a Sphere
The Sky as a Sphere: Days, Seasons and the Phases of the Moon
Every January the Earth swings closest to the Sun. Sunlight is then about 6.9 per cent stronger than in early July. The north is in winter anyway. A 23.44-degree tilt, not distance, makes the seasons.
Pick one
The Moon's orbit tilts only about 5 degrees from the Sun's path. At a new Moon 5 degrees off that path, how close does the Moon come to the Sun?
The Moon's disc is only about 0.52 degrees wide, so 5 degrees is about ten Moon-widths. That is why an eclipse needs a new or full Moon near a node.
The Sky in Six Ideas
Daily turnOne turn against the stars takes 23.93 hours; the sky circles the poles.
Yearly pathThe Sun moves east along the ecliptic, about 1 degree a day.
SeasonsFrom the 23.44-degree tilt, not the distance; perihelion is in early January.
Day lengthSet by latitude and the Sun's declination.
PhasesThe lit half seen from a changing angle (elongation); 29.53-day average cycle.
EclipsesNeed a new or full Moon near a node; the orbit is tilted 5.1 degrees.
One tilted axis (23.44 degrees) and one tilted lunar orbit (5.1 degrees) account for the seasons, the day length and why eclipses are not monthly.
Eratosthenes Measures the Earth: Shadows, Angles and Stated Assumptions
At noon on the June solstice, sundial pointers at Syene cast no shadow and Alexandria's did. From that difference and one distance, Eratosthenes sized the Earth, as Cleomedes tells it. Nobody can now say how long his unit was.
Pick one
Cleomedes gives Eratosthenes 250,000 stades; Strabo and Pliny give 252,000. Converted with the main stade lengths scholars propose, how far off the real circumference is the result?
It depends on the stade, and nobody can now fix its length. Against 40,008 km through the poles, a 157.5 m stade comes out 0.8 to 1.6 per cent low, and a 185 m stade 15.6 to 16.5 per cent high.
From shadow to circumference
1Two shadowsSolstice noon: none at Syene, one fiftieth of a circle at Alexandria.
Models of the Heavens: Ptolemy, Copernicus, Tycho and Kepler
Ptolemy's Earth-centred model hid the proportions of the solar system. His Venus epicycle is 0.719 times the size of its deferent; Venus's orbit is 0.723 times the Earth's. Copernicus read such ratios as distances.
Pick one
Galileo saw Venus round, then half lit. Which arrangements did a round Venus refute: Ptolemy's, Tycho's (Earth still, planets circling the Sun) or Copernicus's?
Ptolemy kept Venus below the Sun, its epicycle centred near the Earth-Sun line. So it never looks more than about half lit. In Tycho's arrangement, as in Copernicus's, Venus circles the Sun and can pass beyond it, where it looks round.
Four models compared
Ptolemy
Earth at rest
Deferents, epicycles, equant
Venus never round
Copernicus
Sun at rest
Retrograde motion as overtaking
Order and scale; tables little better
Tycho
Earth at rest, planets circle the Sun
Same relative motions as Copernicus
Venus can be full
Kepler
Ellipse with the Sun at one focus
Equal areas; T^2 = a^3
Fits Tycho's data; Newton gave the cause
All four save the retrograde loops, and they differ in what is at rest, what they explain and what they predict.
The Seven Liberal Arts, Hands-On: Logic, Proof and the Quadrivium
Euler's polynomial, n squared plus n plus 41, gives a prime for every n from 0 to 39. At n = 40 it gives 1681, which is 41 squared. Testing refutes a rule quickly; it cannot prove it for every n.
Pick one
You want a new square with exactly twice the area of an old one. What should its side be?
Doubling the side quadruples the area; one and a half times the side gives 2.25 times. Cut the old square along its diagonal: the square on that diagonal holds four such halves, the old square two. Both wrong options are the slave boy's guesses in Plato's Meno (82e, 83e); Socrates leads him to the diagonal (85a-b).
Shrink a right triangle to napkin size or stretch it across a football pitch. Keep one angle at 30 degrees, and the side opposite it stays exactly half the hypotenuse. That ratio is the sine of 30 degrees.
Pick one
With no interval stated, how many solutions does sin t = 1/2 have?
One turn holds two, 30 and 150 degrees (pi/6 and 5 pi/6). Adding or removing whole turns gives infinitely many more. In general, t = pi/6 + 2 pi k or 5 pi/6 + 2 pi k, for any integer k.
From geometry to precalculus
1SimilarityEqual angles give proportional sides; this defines sine.
2Pythagorasa^2 + b^2 = c^2, plus its converse and coordinate distance.
3Unit circlePoint at angle t is (cos t, sin t).
4Laws, identitiesLaw of sines, law of cosines, identities and equations.
5PrecalculusInverse functions, conics, vectors, series; then calculus.
Trigonometry grows from similarity, so learn proportion before treating sine as a button.
Knowing the ending makes it look more predictable. Baruch Fischhoff reported in 1975 that people told an outcome rated it likelier than people kept in the dark. The fix: write your prediction down first.
Pick one
A table lamp will not light. Four suspects: the bulb, a dead outlet, the lamp's switch or cord, or a wall switch controlling the outlet. Which first test splits them best?
A working lamp in the same outlet splits the outlet suspects from the lamp suspects in one go. A new bulb, or a look at the switch and cord, tests one suspect. Changing everything at once may fix it without telling you why.
From Surprise to Test
1ObserveState the surprise without explaining it
2HypothesizeGenerate at least three different explanations
3DeduceAsk what else should be true if each is right
4PredictWrite down expectations before running the test
5TestPick the observation that separates hypotheses most
6UpdateRecord the result and revise your beliefs
Commit to predictions before you look, and run the test that best separates the explanations.
A 100 Mb/s link can run flat out and still average 0.6 percent. Over ten seconds, one 0.06-second burst averages the same as a steady 0.6 Mb/s trickle.
Pick one
A 32-bit byte counter wraps to zero after 4,294,967,296 bytes. On a 100 Mb/s link running flat out, how soon does it wrap?
At 12.5 million bytes per second it wraps in about 344 seconds, under six minutes. That is why RFC 2863 provides 64-bit counters for fast interfaces.
Where Assumptions Hide
RatesCounters reset; averages hide bursts.
Packet sizeArithmetic ignores tunnels and the real path.
Recovery timeUnits change the answer; transfer time is a lower bound.
Cost ledgersUnknown is not zero; estimates are not invoices.
UnitskB versus KiB; bits versus bytes.
Before trusting a figure, state the interval, units, denominator and basis behind it.
8 questions from the Astronomy curriculum, Round 1.
Question 1 of 8
What is the obliquity of the ecliptic?
The obliquity is the inclination of the ecliptic to the celestial equator, 23.44 degrees, the same angle as the tilt of the Earth's axis from the perpendicular to its orbit. The 15 degrees an hour is the sky's daily turn, 5.1 degrees is the tilt of the Moon's orbit, and 0.986 degrees is the Sun's daily drift along the ecliptic.
Question 2 of 8
Which ancient author gives the account in which an angle of 7.2 degrees and a distance of 5,000 stades give 250,000 stades?
Cleomedes gives the angle, the distance and the figure of 250,000 stades. Strabo and Pliny report 252,000 stades, and the 400,000 in Aristotle is a figure he cites from earlier mathematicians.
Question 3 of 8
In Ptolemy's model of a planet, what are the epicycle and the deferent?
Ptolemy keeps the Earth at rest and gives each planet a small circle, the epicycle, whose centre rides on a larger circle, the deferent. The point about which the motion is uniform is the equant.
Question 4 of 8
Which fact shows that the Earth's changing distance from the Sun cannot be what causes the seasons?
Perihelion falls in early January, in northern winter. A distance explanation predicts summer in both hemispheres at the nearest point, yet the north is then in winter and the hemispheres have opposite seasons. The first option reverses the dates. The distance does change, from about 147.1 to 152.1 million km. The last option is itself a distance explanation; the lesson gives the causes as the Sun's height and the length of the day.
Question 5 of 8
Why does the shadow angle at Alexandria equal the angle at the Earth's centre between the two cities?
The ray at Syene points to the Earth's centre, and the gnomon's line at Alexandria, extended, also reaches the centre, so it crosses both parallel rays. Equal alternate angles (I.29) make the shadow angle equal the angle at the centre. The Sun is overhead at only one of the cities, and the angle sum of a triangle does not by itself equate the two angles. The circumference is the result, not an input, so using it would be circular.
Question 6 of 8
What did Ptolemy's equant break?
To place the retrograde arcs correctly, Ptolemy made the epicycle's centre sweep equal angles in equal times about a third point, the equant, and not about the deferent's centre. The Earth stayed at rest, the paths stayed circles, and each planet kept its epicycle. It was Copernicus who later replaced the equant with small epicycles.
Question 7 of 8
What causes the phases of the Moon?
The Sun always lights half of the Moon, at every phase, and a phase is how much of that sunlit half faces us. It follows the elongation: 0 degrees is new Moon, 90 degrees a quarter and 180 degrees full Moon. The Moon shines by sunlight, and the Earth's shadow falls on it only in a lunar eclipse, at full Moon.
Question 8 of 8
Which statement about the accuracy of Eratosthenes' result is defensible?
Across the stade values and the two figures of 250,000 and 252,000 stades, the error runs from under 1 per cent to over 16 per cent, and nobody can now fix the unit. The first and third options each pick one disputed stade and treat it as settled, which is why the accuracy claim in kilometres is Aporetic. The Earth's circumferences through the poles and round the equator differ by only 0.17 per cent.