In addition to the simplifying assumptions described in Exercise 547, what are the time intervals that elapse between new moon, first quarter, full moon, last quarter, and the next new moon? (Quarter phases occur when the centers of the Sun, Earth, and Moon form a right-angled triangle, with the right angle at the center of the Moon, which is why we see the terminator as a straight line across the Moon's disk.) The radius of the Earth's orbit and the Moon's orbit should be taken as $150 \cdot 10^{6}$ and $384 \cdot 10^{3} \mathrm{~km}$, respectively.
At the time of the quarter moons, the directions from the Earth (more precisely: from its center) to the Moon and the Sun form an angle $\gamma$ such that $\cos \gamma = r / R$, where $r$ and $R$ are the radii of the Moon's orbit and the Earth's orbit, respectively. From this, $\cos \gamma = 0.00256$, $\gamma = 89^{\circ} 51^{\prime}$.

![](https://cdn.mathpix.com/cropped/2024_05_02_978cc1e633eb9c892d66g-1.jpg?height=456&width=444&top_left_y=276&top_left_x=843)

Geometrically, this can also be stated as follows: the arc of the Moon's orbit between the last quarter and the new moon, and between the new moon and the first quarter, is 9 arcminutes shorter than $90^{\circ}$, while the two arcs extending from the first quarter to the full moon and from the full moon to the last quarter are $90^{\circ} 9^{\prime}$. From a kinematic perspective, it can be said that the $N F H$ angle changes by $\gamma$ from the new moon to the first quarter, giving the Moon an advantage of $\gamma$, or that from the last quarter to the new moon, the Moon has a disadvantage of $\gamma$.

The result of Exercise 547 can also be stated as follows: for the $N F H$ angle to change by $360^{\circ}$, $T = 29$ days 12 hours 44 minutes are required. One quarter of this is 7 days 9 hours 11 minutes, and the fraction of this corresponding to $\gamma$ is the 90th part of the 9/60 part, or $1/600$ part, which (rounded to the nearest whole number) is 18 minutes. Thus, the duration of the "lunar quarters" before and after the new moon is 7 days 8 hours 53 minutes, and the duration of the quarters before and after the full moon is 7 days 9 hours 29 minutes.

András Pinkert (Budapest, Eötvös J. g. II. o. t.)

Remarks: Many solutions involve complex considerations but do not calculate the required times. - Several do not clearly see the relationship between arcminutes and time minutes, which can also arise from notational superficiality. For example, the symbol for 5 time minutes is $5^{m}$, and the symbol for 5 arcminutes is $5_{1}$.