### Thales' Measure of the Sun

June 26, 2023

Just one glance at the periodic table or at one of the many books on my bookshelves reminds me that science progresses by building upon prior scientific discoveries. This idea was expressed in the metaphor written in a 1675 letter of Isaac Newton (1642-1727) to Robert Hooke (1635-1703),
"If I have seen further it is by standing on the sholders of Giants."
As you can see from the word, "sholders," in the quotation, spelling was quite fluid in Newton's time. This metaphor was used several decades before Newton, when Blaise Pascal (1623-1662), in his c. 1647 Preface to the Treatise on Vacuum, wrote about the knowledge we have gained from the ancients:
"...Because having risen to a certain degree where they carried us, the slightest effort makes us rise higher, and with less pain and less glory we find ourselves above them."[1]

Scientists and mathematicians of the past are sometimes honored by the naming of their notable discoveries after them. In recent history, we have the Planck constant, named after German theoretical physicist, Max Planck (1858-1947). Throughout history there have been many such eponymous constants of nature, as detailed on this Wikipedia Page, and there is also a list of physical laws names after scientists.

One of the earliest examples of this in mathematics is the Pythagorean theorem, named after Pythagoras of Samos (c.570-c.495 BC). Prior to that, however, is Thales's theorem, named after Thales of Miletus (c.625-c.546 BC). This theorem, which is perhaps the first mathematical discovery, states that any triangle inscribed in a semicircle with one side as the diameter is a right triangle.

Thales, left, and Pythagoras. (Left image, an 1875 engraving of Thales by Wilhelm Meyer (1844–1944), based on a 4th century bust. Right image of Pythagoras from Wellcome Images, Library reference, ICV No 5031, Photo number, V0004825. Both from Wikimedia Commons. Of course, both are fanciful representations. Click for larger image.)

The most distinguishing aspect of Thales is that he was the first of the ancient Greek philosophers to embrace the idea that things happen, not through the whims of anthropomorphic gods, but according to natural laws. In his quest for first principles without the knowledge of the world as we have it now, Thales believed that water was the ultimate building block of matter. Thales further thought that the Earth floated in a vast body of water, an idea with variations throughout many cultures, including one in which the Earth is supported on the back of a giant turtle.

Thales measured the heights of the Egyptian pyramids using a technique involving similar triangles by what's now called the intercept theorem. He used the shadow of the pyramid caused by the Sun and the shadow of a pole of a known height to get the ratio of the pyramid height to that of the pole. He used the same method to find the distance from shore to ships at sea.

Thales's theorem that any triangle inscribed in a semicircle with one side as the diameter is a right triangle.

I was delighted when I first learned this in high school geometry.

(Created using Inkscape and GIMP.)

Thales was also awarded the equivalent of the Nobel Prize of his time. Bathycles, an Arcadian sculptor and architect, directed in his will that a gold tripod or bowl should be given to him who had done most good by his wisdom. This item was passed among Thales and his fellow Greek sages. Thales initially passed it along, but it was eventually agreed that it should go to Thales. He sent it to the temple of Apollo at Didyma with the dedication, Thales the Milesian, son of Examyas [dedicates this] to Delphinian Apollo after twice winning the prize from all the Greeks.

A recent paper by Jorge Cuadra of the Universidad Adolfo Ibáñez (Viña del Mar, Chile) at the Publications of the Astronomical Society of Japan,[2] and also at arXiv,[3] examines another of Thales' observations; namely the angular size of the Sun. As reported in 2nd and 3rd Century AD manuscripts. albeit far removed from his time, Thales was the first to measure the angular size of the Sun.[2-3]

Cleomedes (c. 2nd Century AD), described how to do this measurement by timing the duration of sunrise, and it's suggested that Thales used the same method.[2-3] Cleomedes gives a size of 0.48°, just 10% smaller than the actual value of 0.5242-0.5422°.[2-3] Thales is reported by Diogenes Laertius (c. 3rd century AD) to have obtained a value of one seven hundred and twentieth part of the solar circle, which is 0.5°. Cuadra argues that uncorrected error arising from the the variation of sunrise duration at different latitudes and at different times of the year is so large as to have prevented Thales from obtaining his roughly accurate value.[2-3]

Illustration of sunrise with solar declination.

The maximum angle of declination, 23°26', occurs at the solstices. It's only at equinox that it's zero, and the Sun moves directly up from the horizon.

(Adapted from fig. 1 of ref. 3.[3] Click for larger image.)

One problem that Thales would have needed to surmount for a sunrise duration measurement was accurate timekeeping. The only clocks available to him were water clocks. I addressed one source of inaccuracy of such clocks in a previous article (Torricelli's law, January 23, 2017). Timekeeping problems aside, Cuadra argues that Thales did not have the ability to make the necessary corrections to sunrise duration for latitude and declination to give his nearly flawless value. Corrections are required if the measurement is not done at the equator during an equinox. equation 4 of his paper illustrates this.[3]

in which Δt is the duration of sunrise in seconds, δ is the solar declination, and φ is the latitude. The range of sunrise duration for Alexandria, where Thales visited, and Miletus, his supposed home, is illustrated in the following graph.

Duration of sunrise, Δt, in seconds, for the latitudes of Miletus (φ = 37.5°), and Alexandria (φ = 31.2°, as a function of solar declination. (Created using Inkscape from data in ref. 3.[3] Click for larger image.)

Thales would have had no idea of the concept of latitude; so, such a correction would be beyond his capability.[3] However, Cleomedes, living seven centuries after Thales, could have applied such a correction.[3] Either Thales used a method different from the duration of sunrise, or his value was corrected before it was mentioned in the 3rd century AD texts.[3]

### References:

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