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The Earth and Its Coordinate System
Published in Terry A. Slocum, Robert B. McMaster, Fritz C. Kessler, Hugh H. Howard, Thematic Cartography and Geovisualization, 2022
Terry A. Slocum, Robert B. McMaster, Fritz C. Kessler, Hugh H. Howard
Describing a point on Earth's surface requires that a location's latitude and longitude be known with respect to an origin. In the case of latitude, the Equator serves as a convenient origin because it divides Earth into two equal halves. Figure 7.2A illustrates lines of latitude, shown as thin red lines, with the Equator represented by the thicker red line. Because lines of latitude are parallel to each other, they are often called parallels. Latitude values are reported in angular measurements of degrees, minutes, and seconds. Similar in concept to units of time, this sexagesimal system has a base unit of 60 (the decimal system's base unit is 10), where each degree is divided into 60 minutes and each minute is divided again into 60 seconds. In this system, the ° symbol denotes the number of degrees, a single quote (′) indicates minutes, and double quotes (″) specify the number of seconds. There are 90° of latitude north and south of the Equator (designated as 0°) for a total of 180° from pole to pole. It is customary to apply the terms North and South to designate latitude locations above or below the Equator. In some cases, plus (+) and minus (–) signs are attached as a prefix to the degree values, indicating latitude locations above and below the Equator, respectively. Thus, the latitude of the Washington Monument can be specified as 38° 53′ 22″ N or +38° 53′ 22″.
Control of the plausibility of measurements
Published in Lucien Wald, Fundamentals of Solar Radiation, 2021
Elevation is often given in m above the mean sea level. Latitudes and longitudes are often given in degree. They may be given in sexagesimal notation, for example, 52° 45ʹ 12ʺ, or decimal one, 52.7533°. As a reminder, converting from a sexagesimal notation (52° 45ʹ 12ʺ) to a decimal is done by first dividing the number of seconds by 60 (12/60 = 0.2). The result is added to the number of minutes; the whole is then divided by 60 (45.2/60 = 0.7533). The result is added to the number of degrees and forms the decimal part, i.e., 52.7533°. Reciprocally, converting an angle in decimal notation (52.7533°) to a sexagesimal one is done by first multiplying the decimal part by 60 (0.753 × 60). It gives the number of minutes in decimal format (45.198ʹ). The decimal part is then multiplied by 60 (0.198 × 60), and this yields the number of seconds, i.e., 52° 45ʹ 12ʺ.
Error and uncertainty
Published in W. Schofield, M. Breach, Engineering Surveying, 2007
The sexagesimal units are used in many parts of the world, including the UK, and measure angles in degrees (°), minutes (′) and seconds (″) of arc, i.e. 1° = 60′1′ = 60′ and an angle is written as, say, 125° 46′ 35′.
Irrationality via base-b representation: an alternative proof of Gauss's lemma
Published in International Journal of Mathematical Education in Science and Technology, 2023
Nowadays, the main base numeration system, as well discussed by Kalapodi (see Kalapodi, 2010) is 10; on the other hand, also the sexagesimal system is still used to measure time and amplitude of angles in mathematics, astronomy and navigations, and in some countries base 12 system has a certain relevance. Although most of the arithmetic properties of base-b operations are very similar (see Hall, 2006), their treatment at the school level often takes place in a detached way (Herman et al., 2011). The evolution of computer science and applied sciences, in general, have given a strong impulse to the study of numbering systems based on arbitrary bases. At different level, our students learn to convert between binary, decimal and hexadecimal representations, they are able to add and subtract numbers in binary (or other) representation. Upon completing these courses, most students are able to perform these procedures proficiently. Are we sure that all of them understand what they are doing (see Herman et al., 2011)?
Henry Bate’s Tabule Machlinenses: the earliest astronomical tables by a Latin author
Published in Annals of Science, 2018
The other preserved version of the Tables of Mechelen survives only in a single manuscript, MS Paris 3091 = Paris, Bibliothèque nationale de France, n.a.l. 3091, fols. 79v–80v (s. XIIIex).42 The mean motion tables are here reduced to elements (i) to (iii) and the sequence of collected years stops at 1320, which is exactly 60 × 20 = 1200 years after the radix. As in Version A, each entry in these tables is divided into four columns, for signs, degrees, minutes, and seconds. In the case of some extended-year tables, however, the final line has additional numbers in the right margin that serve to render the 20-year increment of the corresponding mean motion more precise. In the case of the Sun, Moon, and Mars, the scribe merely added a number for sexagesimal thirds, which could have easily been inferred from the 1200-year increment in the collected-year table (since 1200 = 60 × 20). A more interesting case are the tables for Saturn and Venus, where the 20-year values receive an extension to the fourth sexagesimal place. This presumably reflects some deeper knowledge about the underlying parameter and hence points to Henry Bate as the author of these additions.
On the mother bodies of steady polygonal uniform vortices. Part I: numerical experiments
Published in Geophysical & Astrophysical Fluid Dynamics, 2022
As shown in figures 5(a) and 6(a) for triangular and square vortices, respectively, the real part is an increasing function of along the cut. On the contrary, figures 5(b) and 6(b) show that the amplitude of the jump monotonically decreases, up to vanish in correspondence to the branch point (). The behaviour of in a neighbourhood of the branch point is well described by the series (22), as shown in the same figures, where the partial sums of order 5 of the above series (dashed lines) are superimposed to the curves (solid) resulting from the numerical integration of equation (23). It can be easily checked that, inside the convergence interval specified in the previous section, the two kinds of curves are not distinguishable, and this occurs for both real and imaginary parts. In particular, the series (22) gives in any sufficiently close to the branch point () the derivative , the real part of which approaches from the left and from the right (remember that ), while the imaginary one diverges to from the left and vanishes from the right. As a consequence, the tangent to the curve as has the finite slope , while it is vertical as : in (on the branch point) the curve has a corner point. Some values of the angle γ (sexagesimal degrees) are listed in tables A1 and A2 for triangular and square vortices, respectively.