In 1614 Johann Georg Locher, a student of the Jesuit astronomer Fr. Christoph Scheiner, published a short book called Disquisitiones mathematicae, de controversiis et novitatibus astronomicis—that is, Mathematical Disquisitions, Concerning Astronomical Controversies and Novelties. Among other things, the book discussed the phases of Venus and the satellites of Jupiter, all recently discovered by Galileo using a telescope. Toward the end of the book, Locher engaged in an interesting exercise in astronomical reasoning. He presented to his readers the figure below.

And, regarding this figure, he wrote—

Vnam hactenus Lunam agnouimus circa terram, quam oculus A in terra positus libere conspexit, modo silentem in B; modo dimidiam in C; alias plenam in D; alias curtatam in E.... At vero, post repertum Oculum Astronomicum, tubum inquam Opticum, plures sese aperuerunt nobis Lunae. Quarum praecipua videtur esse Venus; ea enim in tubum GH ex I delapsa, oculo A occurrit falcata, dum puncto M Augis opposito vicina agit: & vero in K Auge plena radiat; semiplena in L elongatione media; in M inanis omnino apparet. Et sic Lunam alteram habemus circa Solem veluti annuam.... Alias etiam quatuor Lunulas habemns Iouias, nec oculo tamen, neque tubo, sed ratione detectas. Cum enim ab umbra Iouis eclipsentur, obtutus eosdem aduertent Ioui, quas Luna aut Venus terrae. Vnde oculus in Ioue constitutus, videret eam quae ad O corniculatam; quae ad P plenam; quae ad Q dimidiam aut gibbam; quae ad R vacuam: Lunularum porro harum Iouialium alia est sesquiduana, alia triduana, &c. alia octiduana fere; alia decimoquartana. Quod si homo in Saturno constitutus esset, aspectaret is omnes omnino planetas instar Lunarum.... Up until now we have acknowledged one moon, circling around Earth, which the unaided eye A [in the Figure] positioned on Earth sees dark when at B, half when at C, full at D, and short of full at E.... But truly with the invention of the optic tube [i.e. the telescope] or Astronomical Eye, more “moons” have revealed themselves to us. Venus is one especially. Indeed, Venus viewed by the eye A, through the tube GH, appears crescent when at I, grows as it moves away from M, becomes full at K, and half-full at L (when most separated from the sun). At M it appears entirely dark. And thus we have circling the sun a second “moon” with an approximately annual cycle.... We now have an additional four other little Jovian moons, detected not by the eye, nor by the tube, but by reason. Since they are in fact eclipsed by the shadow of Jupiter, they will present the same appearances to Jupiter that the moon or Venus present to Earth. And so an eye positioned on Jupiter might see a satellite crescent shaped when at O, full when at P, half at Q, and dark at R. Moreover, one of these little moons completes a cycle in about one and one-half days, another in about three days, the next in about eight days, and the last in about seventeen days. If a man were placed on Saturn, he might observe all the planets to look like the Moon....

Locher reasoned that the satellites of Jupiter must show phases—crescent, full, etc. Why? We see with our eyes that our Moon has phases. Our Moon’s phases occur because our Moon does not itself emit light: it is a dark body, illuminated by the sun. That half of the globe of our Moon that faces the sun is bright, the other half is dark. The relative positions of our Moon, the Earth, and the Sun determine how much of that illuminated half of our Moon we will see, and thus determine the phase of our Moon. For example, if our Moon is opposite the Sun in the sky, so that Earth lies between it and the Sun (at D in Locher's figure), we see only the illuminated half of our Moon, and none of the unlit half, so we see a Full phase. Moreover, we see with the telescope that Venus has phases, just like our Moon. Venus then is also a dark body, illuminated by the Sun.

Locher observed that the Jovian satellites could not be seen when they were in Jupiter’s shadow. If they emitted their own light they would always be visible, whether they were in Jupiter’s shadow or not. Thus, he reasoned, they must be dark bodies that do not emit their own light—dark bodies like our Moon. And, they circle Jupiter like our Moon circles Earth. Thus they must have the same appearance as seen from Jupiter that our Moon has as seen from Earth. In other words, as seen from Jupiter, they must show phases. Logic and geometry (that is, mathematics) say they must.

Locher could not detect these phases with his eye; indeed, the Jovian satellites are not visible to the unaided eye. Locher could not detect these phases with the telescope; all he could see of these satellites were dots of light. Even telescopes far better than Locher’s cannot directly reveal whether Jupiter’s moons show phases, because Earth, being fairly close to the sun compared to Jupiter, cannot be in a position to see the unlit halves of the Jovian satellites. It is reason that detects the phases of the Jovian satellites, says Locher, not direct observation with the eye or telescope.

We should note that reason does not absolutely prove that Jupiter’s satellites are dark bodies illuminated by the Sun and that therefore they show phases. We could theorize that perhaps each Jovian satellite does emit its own light, like a stupendous light bulb. And perhaps each satellite’s light is controlled by an elf who chooses to switch off that satellite’s light every time the satellite passes into Jupiter’s shadow (we can speculate that this occurs because the elves like the dark and can only get complete dark when in Jupiter’s shadow and thus shaded from the Sun). Were the “elf hypothesis” correct, then the glowing satellites would absolutely not show phases as seen from Jupiter.

But, the elf hypothesis is not correct. NASA has sent robotic space probes out to Jupiter. Those probes have been in a position to see the unlit halves of the Jovian satellites, and the cameras on those probes have seen directly that Jupiter’s moons are dark bodies, illuminated by the Sun. The probes have proven that the moons show phases, like Locher said. That’s one more victory for reason, logic, and mathematics, and one more loss for invoking elves. Of course, no one thinks of “proving that Jupiter’s satellites are moons and not elf-controlled light bulbs” as being a big astronomical discovery made by NASA’s probes, because the rational argument for them being moons and not elf-bulbs was so compelling.

Invoking elves may seem silly, but the conclusion Fr. Scheiner’s student reached in 1614—that Jupiter’s moons show phases, a conclusion borne out over 350 years later by NASA probes—is much like many other conclusions about the universe that astronomers arrive at. These conclusions—such as the existence of dark energy, black holes, “planet nine”, or the Big Bang—are reached through reasoning, logic, and mathematics, sort of like we use reason to figure out the digits of pi. These conculsions are not proven absolutely. Some may never be proven absolutely through any direct observation. But, they make a lot of sense.