Tuesday, January 06, 2015

rambam's estimate of the size of the moon

Rambam writes (Yesodei haTorah 3:8):

הארץ גדולה מן הירח כמו ארבעים פעמים

I am curious if anyone has an idea where the Rambam got this from.  The Rambam writes in Hil Kiddush haChodesh (ch 17) that the ancient Greeks figured out the math and astronomy needed to calculate our calendar.  It seems that the Greeks already knew that the moon was about ¼ the size of the earth, so where did the Rambam get the figure 1/40 from?  (I am not concerned with the number being off – I am just looking for a source.) 

I'm tempted to suggest that there is a printer’s error here and the Rambam should read "k'mo arba" instead of "k'mo arba'im"  (I haven’t checked the Frankel edition to see if there are other girsa’os).



  1. The VOLUME of the moon is about 1/45 of the volume of the Earth. (I found this answer in the sefer חזון שמים)

    1. Sorry, I'm not convinced. The Rambam in that same halacha writes
      וְהַשֶּׁמֶשׁ גְּדוֹלָה מִן הָאָרֶץ כְּמוֹ מֵאָה וְשִׁבְעִים פְּעָמִים
      The volume of the sun is over a million times that of the earth.

    2. See my comments below. Rambam presumably used the best estimates for sun radius that were known in his day, which we now know were 20 times too small. Since volume is proportional to the cube of a sphere's radius, Rambam's estimate of the sun's volume was about 8000 times smaller than our modern estimate. That's why he thought the sun was only 170 times larger in volume than the earth, instead of over a million times larger in volume as we believe today.

    3. Aristarchus, who lived a century after Aristo, used lunar eclipses (the earth's shadow is roughly the size of the moon) and angles to get a ratio of diameters of 1:2.86. (The moon really is 1:3.6 of the earth in diameter.) The Greeks didn't have 1:4, and his man Aristo didn't have any estimate.

      Newton computed the mass of the moon to be 1:40 the mass of the earth. (Principia Book III Prop. 37, Cor. IV) In reality, it's more like 1:81, but perhaps the Rambam used a similar estimation as Newton, leading to the same mistake.

      As for the Rambam's estimate for the sun... Unless you think Aristotle had ruach haqodesh, I wouldn't ascribe Divine Knowledge to much of the first 5 chapters of Yesodei haTorah. It's easier to compute data about the moon. His being wrong about the harder problem doesn't necessitate his being wrong about the easier one. But I think he was simply wrong.

    4. I think Rambam probably had access to Ptolemy and his updates/improvements to Arisrarchus. Ptolemy's work was known as Almagest, and was translated into Arabic by the 9th century. Rambam read lots of classic works besides Aristotle, pretty much everything that was translated into Arabic. Anyway, if so, he had the (diameter) ratios in my comments below, which agree pretty well with the (volume) ratios claimed by Rambam.

      And yes, I agree the science of that day was wrong. But I think Rambam was using the best science of his time.

  2. See here: http://www.universetoday.com/20489/moon-compared-to-earth/

  3. So the Rambam simply took the Greeks' value and converted it from diameter to volume.

  4. I believe dlz is correct, Rambam was speaking about volume, not radius or diameter.

    Volume of a sphere is proportional to the cube of its radius (or diameter).

    I suspect Rambam used Ptolemy's estimates for the relative diameters of earth, moon and and sun. Ptolemy was the top authority on astronomy from the end of antiquity until the Copernican revolution. Maybe there were refinements, but Rambam almost surely used Ptolemy or else something very close. Here are the Ptolemy ratios:

    diameter of earth:moon = 3.42:1
    diameter of sun:earth = 5.5:1
    so diameter of sun:moon = 18.84:1

    Taking the cubes of those ratios yields
    - relative volume of earth:moon = 40.002, indeed very close to 40
    - relative volume of sun:earth = 166, also close to Rambam's claim of roughly 170
    - relative volume of sun:moon = 6687:1, pretty close to Rambam's claim of about 6800

    I found these Ptolemy diameter estimates at:

    1. Slight correction: volume of sphere is proportional to cube of its radius (not diameter). But still, the Ptolemy ratios for relative diameter of the earth/sun/moon are exactly the same as their relative radius ratios. So all of the remaining logic works out the same.

    2. Given Ptolmey's estimate of the sun's diameter, I think you/dlz are right.

  5. The Rambam, in Hilchos Shavuos 5:2, kind of tells us where he got it from:
    "דבר ידוע אצל החכמים בעלי שכל ומדע שהשמש גדולה מן הארץ מאה ושבעים פעמים"
    Granted, though, there he only mentions the sun, not the moon.

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  7. First, my opinion, is that if Rambam has mentioned those 2 ratio sun/earth and earth/moon and then do a trivial multiplication if each after is that both ratio are not so trivial, non radius, volume, distance, mass, etc are mentioned. Then each ratio should be consistant with multiple other vallues and évents: solair and moon éclipse .

    Below (Sorry In french because i forwared a copy/pats to my rabbi waiting for its opinion) is my opinion to catch those 2 ratios and chalenge with you.

    A) Une explication plausible du rapport soleil/terre near 167

    1) Le rayon soleil est de ~700000km.
    2) La couronne solaire visible uniquement lors des éclipses totales de soleil est estimé aujourd'hui à 1/2 rayon soleil, généralement sphérique.
    3) le rayon terrestre est ~6400 km.
    --> Le ratio ((1) + (2))/(3) est proche de ~164.

    Cependant la taille de la couronne est une estimation. Et je défie quiconque de dire que ce 'est pas 0.52 au lieu de 0.5...Donc rien ne prouve today que cette taille soit très légèrement supérieure pour avoir un ratio 167...
    However for Rambam, this ratio is exactly 166 + 3/8 (or 5.5)^3. But clearly i don't have this solution.... Not for the moment, missing for info.

    B) Ratio terre / lune proche de 40
    1) distance minimum terre/lune (perhilée) =376400km (voir Wikipédia)
    2) lors d'une éclipse totale de lune, le cône d'ombre de la terre dans lequel traverse la lune est proche de 9300km (approximation diamètre terre ~ 12800 km moins diamètre lune ~3500km; approximation valable car l'angle de visibilité de lune ~ soleil ~ 0.5° sur la terre).
    --> Ratio (1)/(2) est très proche de 40...

  8. Letter from the Rebbe, Menachem M. S.

    B”H, Rosh Chodesh Tammuz, 5722

    Peace and Blessing!

    The statement in Sha’ar HaYichud V’haEmunah (chapter 7) that the sun “is approximately 167 times the size of Earth” is based on the Rambam’s ruling in Hilchot Yesodei HaTorah 3:8. (In his introduction to Perush HaMishnayot he gives a more precise measurement of 166 3/8 times the size.)

    It seems that people have challenged you with the alleged view espoused by modern astronomers, according to which the sun is four to five million times the size of earth, or more.

    Just as with all other “scientific” questions of this type, the questioner is obviously ignorant about science as well!

    Astronomy’s assertion that the sun is 4 to 5 million times the size of the Earth is with regard to the volume, while Rambam is discussing the diameter. True, modern science’s estimate of the diameter of the sun is right now about 110 times the size of Earth’s, rather than 170. However, that measurement of the sun’s diameter takes into account only certain layers of the sun, and not all of them. The outer layers (according to today’s astronomers) change (in a cycle of constriction and expansion) greatly from time to time; also, not all of them can be seen— some can only be measured by their effects, etc. Thus, it is hard to measure them.

    Similarly, they don’t take into account the protuberance. At times, this protuberance was measured as having reached such sizes, as to take the sun’s diameter to more than 170 times the Earth’s diameter, after which it again constricts, etc. This is not the place to write at greater length30.