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The definition of time and different time systems
Published in Lucien Wald, Fundamentals of Solar Radiation, 2021
The definition of year is not unique: sidereal, Julian, tropical year…; but overall, it is about 365.25 days. There is therefore a discrepancy between this duration and a regular year of 365 days, used in the Gregorian calendar, that of every day in very many countries of the world, and used in this book. The difference is made up every 4 years, by adding a day (29 February). This 366-day year is called a leap year. Of course, when it comes to timing and time, things are not that simple. The years are leap years only if the year is divisible by 4 and not divisible by 100, or if it is divisible by 400. Note that in climatology, data of 29 February are excluded from time series in order to have the same number of days for each year.
Discussion on the importance of mathematical models
Published in Jun Wu, Rachel Wu, Yuxi Candice Wang, The Beauty of Mathematics in Computer Science, 2018
The high precision of Ptolemy's model amazed later generations of scientists. Today, even with the help of computers, it is still a big challenge to solve equations of 40 inscribed circles. Every time I consider this, I admire Ptolemy from the bottom of my heart. The Julian Calendar matches Ptolemy's calculations, i.e., every year has 365 days, and every four years we have a leap year which adds an extra day. Over the span of 1,500 years, people relied on Ptolemy's calculations for the agricultural calendar. However, 1500 years later, the accumulated error of Ptolemy's proposed path of the Sun is equal to about 10 extra days. Due to this error of 10 days, European farmers were off by almost one whole solar period, which had a huge impact on agriculture. In 1582, Pope Gregory XIII took off 10 days from the Julian Calendar. Furthermore, the leap year following every century was changed into a regular year, but the leap year after every 400 years remained a leap year. This is the solar calendar that we still use today, and is almost completely free of error. In order to commemorate Pope Gregory XIII, this calendar is now called the Gregorian Calendar.
The Earth in Space
Published in Aurèle Parriaux, Geology, 2018
The Earth takes 365.256 days to revolve around the sun, which necessitates one leap year every four years. The length of the year has probably not changed significantly during the history of the Earth. This is not true for Earth’s rotation on its axis. Today, the day is 23.934 hours long. The study of coral growth from its appearance in the mid Paleozoic until today shows that the number of daily cycles in the seasonal cycle of older fossils (Fig. 2.9) is greater than that of younger fossils. This means that Earth’s rotation has slowed down over the course of geologic time, as a result of braking due to energy consumption by the tidal deformation of Earth (especially by the moon’s attraction, Chap. 9).
Development of a temperature prediction model for asphalt pavements considering air temperature data of preceding hours
Published in International Journal of Pavement Engineering, 2022
Ashish Walia, Rajat Rastogi, Praveen Kumar, S. S. Jain
A correlation study was carried out to determine the correlation coefficients between the observed pavement temperatures and average air temperature computed for the ‘x’ preceding hours, resulting in 72 values for each observation location. The hourly temperature data used were for an entire year (1 August 2019, to 31 July 2020), resulting in a total of 8784 values (i.e. 366 days × 24 h) across different depths in the AC layer. It should be noted that the year 2020 was a leap year, and hence, the total number of days was 366. Considering that x = 0 represents current temperature, 72 correlation coefficient values were obtained for the relation between pavement temperature and the average air temperature (up to 71 preceding hours). These correlation coefficients were calculated at different depths for all four sections and are plotted, as shown in Figure 4.
The mathematics of leap years: sophistication versus content in mathematics education
Published in International Journal of Mathematical Education in Science and Technology, 2022
Yip Cheung Chan, Kam Moon Pang, Kenneth Young
A leap year is a calendar year with 366 days rather than 365. The basic regularity, already found in the Julian calendar, is familiar: (Rule 1) every year divisible by 4 (e.g. 2012, 2016) is a leap year. The refinements (Swinburne University of Technology, n.d.), adopted in the Gregorian calendar since 1582, are less known and sometimes omitted (Collins, n.d.): (Rule 2) every year divisible by 100 is not a leap year, except that (Rule 3) every year divisible by 400 is a leap year. Why are Rule 2 and Rule 3 needed? Where do the numbers 100 and 400 come from? Are there (in principle) better rules? And what does ‘better’ mean? Thinking through these issues will lead to (a) an interesting mathematical problem, as yet not precisely defined, that can be approached at different levels of precision, generality and rigour, and (b) broader reflection on pedagogy, as explained below.
A mestizo cosmographer in the New Kingdom of Granada: astronomy and chronology in Sánchez de Cozar Guanientá’s Tratado (c.1696)
Published in Annals of Science, 2021
Sergio H. Orozco-Echeverri, Sebastián Molina-Betancur
The measure of the great circulation of the Sun rests on the exact duration of the solar year which, in Sánchez’s view, requires corrections. The first step in the argument is evaluating previous calculations as they appeared in Zamorano’s Cronología.118 While Hipparchus and Ptolemy defined the duration of the solar year in 365 days, 5 h, 55 min and 12 s, Al-Battani reckoned it in 365 days, 5 h, 46 min and 24 s. By the time of the Gregorian reform, ‘Copernio (sic) and the Prutenic tables of our King Alfonso (sic)’ gave the same values as Hipparchus and Ptolemy.119 As a result, ‘astrologers’ working on the Gregorian reform defined the mean year in 365 days, 5 h, 49 min and 16 s, and therefore, they introduced an inaccurate rule for leap years.120 However, over hundred years after the reform was introduced in 1582, Sánchez noticed that ‘the aspects and conjunctions and the full moons’ do not match the Sun. Thus, ‘our King Alfonso X missed 44 s to determine the entire quantity [of the year]’.121 The correction consists in evaluating the mean annual motion of the sun through the zodiac, measured from the point of the vernal equinox — the first degree of Aries. The path of the Sun calculated by the Gregorian reformers fell behind the actual distance which the Sun travels when it is about to complete the period of four years before the introduction of a leap year: As the natural motion of the sun from west to east through the zodiac (which the astrologers call mean motion) takes 59 min, 8 s, and 14 thirds of degree, in one year [the sun] travels 11 signs, 29 degrees, 45 min, 5 s and 10 thirds (falling behind the first degree of Aries by 14 min, 54 s, and 50 thirds). Consequently, given that in 4 years [the Sun] would have travelled 47 signs, 29 degrees, no minutes, 20 s, and 40 thirds (falling behind the first degree of Aries by 59 min, 39 s, and 20 thirds, and given that because of this delay a leap year should be simulated), the quantity of the year should be no more and no less than 365 days, 5 h, and 50 min.122Sánchez does not provide the source of these numbers. In any case, his method for determining the solar year consisted in the multiplication of the distance travelled by the Sun in one day by the 365 days of the year, and then this result by four years, corresponding to the period in which a leap year should take place. Here Sánchez found the missing seconds that, once added, would balance the calendar with the solar year and ultimately would re-establish the date of the vernal equinox.