# Origin of arcminutes, arcseconds, “arcthirds,” “arcfourths,” etc

This section of a Wikipedia article says

[Modern time and angle notation] contrasts with the numbers used by Hellenistic and Renaissance astronomers, who used thirds, fourths, etc. for finer increments. Where we might write 10°25'59.392", they would have written 10°25′59″23‴31⁗12′′′′′ or 10025I59II23III31IV12V.
1. Are these claims correct? and if so, according to what secondary source?
1. Were Roman numerals used in superscript with this meaning?
2. Were divisions beyond the arcsecond used?
2. Is the origin of the prime (typographic symbol), and double and triple prime, in the Roman numerals I, II, and III?
3. Is the origin of the degree symbol, similarly, in a superscript zero?

See F.Cajori, A History of Mathematical Notations: Vol. II, para 511 : Origin of the modern symbols for degrees, minutes, and seconds :

Signs resembling those now in use are found in the Syntaxis (Almagest) of Ptolemy (ca. AD 100 – ca. 170), where the Babylonian sexagesimal fractions are used in astronomical calculations. The units were sometimes called μοῖραι and frequently denoted by the abbreviation $$\mu^°$$. The first sixtieths or minutes were marked with one accent, the second sixtieths with two accents. [...] From these facts it would seem that our signs $$^°, ', ''$$ for degrees, minutes, and seconds were of Greek origin. But it is difficult to uphold this view, especially for the sign $$^°$$ for "degrees." Such a line of descent has not been established.

See an example of medieval astronomical table : the so-called Alfonsine tables, compiled in Toledo, Spain, and containing astronomical data starting on January 1, 1252, the date of the coronation of the King Alfonso X of Castile.

During Middle Ages and Renaissance [see Cajori, para 512]

the names signa, gradus, minutae, secundae, etc., were used with several abbreviations; the more common ones are Sig., Gr., Min., Sec.

Ulricus Regius in his Epitome of 1536 refers to the Alfonsine tables and gives the signs

$$T, s, g, \overline m, \overline s, \overline t, \overline {qr}$$. [Denominator minutorum est unitas. Secundorum binarius, Tertiorum ternarius, Etc.]

In 1540 Gemma Frisius wrote :

"$$\text{Integr. Mi. 2. 3. 4.}$$

$$\ \ \ \ \ \ \text{36. 30. 24 50 15}$$"

for our modern $$36^°30'24''50'''15^{iv}$$. This is the first modern appearance that I have found of $$°$$ for integra or "degrees."

In 1571 Erasmus Reinhold gave an elaborate explanation of sexagesimal fractions as applied to angular measure and wrote "$$62° 54' 18''$$." This notation was adopted by Tycho Brahe who in his comments of 1573 on his Nova Stella writes $$75° 5'$$, etc.

In conclusion : yes, roman numerals were adopted as superscript for the fractions of degree.

But the "scientific notation" for fractions used in Astronomy was not used in "ordinary" mathematics.

See e.g. Ancient roman fractions :

The Romans did not use numerals to indicate fractions, but instead used words to indicate parts of a whole. A unit of weight was the as and the uncia (from which we have the word "ounce") was a twelfth part of the as.

See e.g. the term deunx for the fraction $$\dfrac {11} {12}$$ (from de uncia, i.e. $$\dfrac {1}{12}$$ taken away).