According to wikepedia, Hypsicles (Greek: Ὑψικλῆς; c. 190 – c. 120 BCE) was an ancient Greek mathematician and astronomer in Alexandria, known for authoring On Ascensions (Ἀναφορικός) and the Book XIV of Euclid’s Elements. In his book On Ascensions (On Rising Times) he divides the circle into 360 equal parts, which we now call degrees, and he “proves a number of propositions on arithmetical progressions and uses the results to calculate approximate values for the times required for the signs of the zodiac to rise above the horizon.”
Knowing the rising times of star groups was useful for telling time at night. Hypsicles in the 2nd century BCE applied a Babylonian method of arithmetic progression to estimate the actual rising times. Later, with the development of trigonometry, an exact measurement of rising times became possible. By the 2nd century CE Ptolemy had published tables of rising times in his texts. Here is an image of a table of ascenional times from a 9th century CE edition of Ptolemy’s Almagest:
Using spherical trigonomety, mathematicians of Ptolemy’s era were able to calculated actual acensional times of zodiac signs.
In a recent post on FaceBook astrologer Margherita Fiorello presented a diagram outlining the ascencional times of the zodiacal signs proposed by Hypsicles on the basis of an arithmetic progression rather than direct observation of the rising of the signs. Ideally it would take each sign 2 hours to rise but due to the obliquity the ecliptic and the location of the observer, some signs take more than 2 hours to rise and other signs take less than 2 hours. Here is a copy of Margherita Fiorello’s redrawing of a diagram of Hypsicles model which ancient astrologers used to estimate when certain primary directions would manifest in the lives of their clients:
For comparison, here is a Hellenisitic chart, from Delphic Oracle software, with ascensional times in the center for Cancer rising on the day of the summer solstice of 2021 in Alexandria.
An astrologer looking at the above chart might wonder when in the life of an individual born at this time Venus in the 1st house would rise to the horizon by primary direction. An easy estimate can be calculated by consider that Venus is about 22.5 degrees from the 0 Cancer 9 Ascendant, which is 22.5/30 = 0.75 of the way through Cancer. Since Cancer has an ascensional time of 34.56, we multiple that figure by the 0.75 and estimate that Venus will cross the Asc by primary direction when the person is about 25.9 years old. Rounding the 25.9 to 26 years and adding 26 to the current year 2021, we might expect the year 2047 to be a particularly Venusian one, possibly a good year for a wedding.