^{1}

^{2}

^{2}

^{3}

The zonal averages of temperature (the so-called normal temperatures) for numerous parallels of latitude published between 1852 and 1913 by Dove, Forbes, Ferrel, Spitaler, Batchelder, Arrhenius, von Bezold, Hopfner, von Hann, and B örnstein were used to quantify the global (spherical) and spheroidal mean near-surface temperature of the terrestrial atmosphere. Only the datasets of Dove and Forbes published in the 1850s provided global averages below 〈T 〉=14 °C, mainly due to the poor coverage of the Southern Hemisphere by observations during that time. The global averages derived from the distributions of normal temperatures published between 1877 and 1913 ranged from 〈T 〉=14.0 °C (Batchelder) to 〈T 〉=15.1 °C (Ferrel). The differences between the global and the spheroidal mean near-surface air temperature are marginal. To examine the uncertainty due to interannual variability and different years considered in the historic zonal mean temperature distributions, the historical normal temperatures were perturbed within ±2σ to obtain ensembles of 50 realizations for each dataset. Numerical integrations of the perturbed distributions indicate uncertainties in the global averages in the range of ±0.3 °C to ±0.6 °C and depended on the number of available normal temperatures. Compared to our results, the global mean temperature of 〈T 〉=15.0 °C published by von Hann in 1897 and von Bezold in 1901 and 1906 is notably too high, while 〈T 〉=14.4 °C published by von Hann in 1908 seems to be more adequate within the range of uncertainty. The HadCRUT4 record provided 〈T 〉 ≌ 13.7 °C for 1851-1880 and 〈T 〉=13.6 °C for 1881-1910. The Berkeley record provided 〈T 〉=13.6 °C and 〈T 〉 ≌ 13.5 °C for these periods, respectively. The NASA GISS record yielded 〈T 〉=13.6 °C for 1881-1910 as well. These results are notably lower than those based on the historic zonal means. For 1991-2018, the HadCRUT4, Berkeley, and NASA GISS records provided 〈T 〉=14.4 °C, 〈T 〉=14.5 °C, and 〈T 〉=14.5 °C, respectively. The comparison of the 1991-2018 globally averaged near-surface temperature with those derived from distributions of zonal temperature averages for numerous parallels of latitude suggests no change for the past 100 years.

Analyses of global surface-temperature change have been routinely carried out by several groups, including the NASA Goddard Institute for Space Studies (NASA GISS), the NOAA National Climatic Data Center (NCDC), and a joint effort of the UKMet Office Hadley Centre and the University of East Anglia Climatic Research Unit (HadCRU) [

( 〈 T 〉 S H = 13.0 ˚ C and 〈 T 〉 N H = 14.3 ˚ C ) for 1881-1910. The Berkeley record provided 〈 T 〉 = 13.6 ˚ C and

〈 T 〉 = 13.5 ˚ C for these periods, respectively. The NASA GISS records yielded 〈 T 〉 = 13.6 ˚ C

( 〈 T 〉 S H = 13.0 ˚ C and 〈 T 〉 N H = 14.2 ˚ C ) for 1881-1910. When we considered 1991-2018, the HadCrut4 re-

cord yielded 〈 T 〉 = 14.4 ˚ C ( 〈 T 〉 S H = 13.7 ˚ C and 〈 T 〉 N H = 15.2 ˚ C ), the Berkeley record provided

〈 T 〉 = 14.5 ˚ C , andthe NASA GISS record provides 〈 T 〉 = 14.5 ˚ C ( 〈 T 〉 S H = 13.7 ˚ C and 〈 T 〉 N H = 15.2 ˚ C ). Thus, these results suggest an increase in the globally averaged near-surface temperature during the past 100 years of 0.8˚C, 1.0˚C, and 0.9˚C, respectively.

In 1900, von Bezold [

Ferrel [

A slight improvement of Dove’s data was derived by Forbes [^{th} and the 10^{th} degree of latitude, respectively. In contrast to this, von Bezold [

Von Bezold [

In 1906, von Bezold [

Von Hann [

Author | Year of publication | NH (˚C) | SH (˚C) | Earth* (˚C) |
---|---|---|---|---|

Dove [ 19 ] | 1852 | 15.5 | - | - |

Schoch [ 20 ] | 1856 | 15.1 | 14.9 | 15.0 |

Satorius von Waltershausen [ 21 ] | 1865 | - | 15.8 | - |

Ferrel [ 12 ] | 1877 | 15.3 | 16.0 | 15.7 |

Spitaler [ 10 ] | 1885 | 15.4 | 14.8 | 15.1 |

von Hann [ 28 ] | 1882 | - | 15.4 | - |

von Hann [ 15 , 16 ] | 1897/1903 | - | 14.7 | - |

*) Based on Equation (9).

Latitude in ˚ | Author and data set (temperatures in ˚C) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Dove Do1852 | Forbes Fo1859 | Ferrel Fe1877 | Spitaler Sp1885 | Batchelder Ba1894 | Arrhenius A1896-1 | Arrhenius* A1896-2 | Hopfner Ho1906-1 | Hopfner Ho1906-2 | Defant and Obst De1923 | Von Hann-Süring vHS1939 | Sellers Se1965 | |

90 N | −16.5 | −16.5 | - | −20 | - | - | - | - | - | −22.7 | −22.7 | - |

85 | - | - | - | - | - | - | - | - | - | - | −21.2 | −23.5 |

80 | −14 | −14 | −15.3 | −16.5 | −16.9 | - | - | −16.1 | −16.4 | −17.1 | −17.2 | - |

75 | - | - | - | −13.3 | - | - | - | - | - | - | −14.7 | −15.8 |

70 | −8.9 | −8.7 | −9.8 | −9.9 | −10.1 | - | - | −10.1 | −9.6 | −10.7 | −10.7 | - |

65 | - | - | - | −4.3 | - | −5.5 | −7 | - | - | - | −5.8 | −7.1 |

60 | −1 | −1.2 | −1.5 | −0.8 | −1.3 | - | - | −1.1 | −0.8 | −1.1 | −1.1 | - |

55 | - | - | - | 2.3 | - | 2.5 | 1.2 | - | - | - | 2.3 | 0.6 |

50 | 5.4 | 5.8 | 6.3 | 5.6 | 5.8 | - | - | 5.8 | 6.2 | 5.8 | 5.8 | - |

45 | - | - | - | 9.6 | - | 10.3 | 8.7 | - | - | - | 9.8 | 7.6 |

40 | 13.6 | 13.6 | 13.6 | 14 | 14 | - | - | 14.6 | 14.5 | 14.1 | 14.1 | - |

35 | - | - | - | 17.1 | - | 17.5 | 15.3 | - | - | - | 17.2 | 14.1 |

30 | 21 | 21 | 19.8 | 20.3 | 20.2 | - | - | 20.8 | 20.7 | 20.4 | 20.4 | - |

25 | - | - | - | 23.7 | - | 23.1 | 21.9 | - | - | - | 23.6 | 20.5 |

20 | 25.2 | 25.3 | 25.3 | 25.6 | 24.9 | - | - | 25.3 | 25.2 | 25.3 | 25.3 | - |

15 | - | - | - | 26.3 | - | 26.2 | 25.4 | - | - | - | 26.3 | 25.2 |

10 | 26.6 | 26.6 | 27.2 | 26.4 | 27.1 | - | - | 26.7 | 26.8 | 26.8 | 26.7 | - |

5 | - | - | - | 26.1 | - | 26.7 | 25.5 | - | - | - | 26.4 | 25.6 |

0 | 26.5 | 26.5 | 26.7 | 25.9 | 26.6 | - | - | 26.4 | 26.3 | 26.3 | 26.2 | - |

−5 | - | - | - | 25.5 | - | 26.1 | 25.1 | - | - | - | 25.8 | 24.9 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

−10 | 25.5 | 25.6 | 25.9 | 25 | 25.7 | - | - | 25.6 | 25.4 | 25.5 | 25.3 | - |

−15 | - | - | - | 24.2 | - | 24.3 | 23.2 | - | - | - | 24.4 | 23.4 |

−20 | 23.4 | 23.4 | 23.7 | 22.7 | 23.3 | - | - | 23 | 22.9 | 23 | 22.9 | - |

−25 | - | - | - | 20.9 | - | 20.6 | 19.7 | - | - | - | 20.9 | 18.9 |

−30 | 19.4 | 19.4 | 19.3 | 18.5 | 18.3 | - | - | 18 | 18.3 | 18.4 | 18.4 | - |

−35 | - | - | - | 15.2 | - | 14.8 | 14.5 | - | - | - | 15.2 | 13.6 |

−40 | 12.5 | 12.6 | 14.4 | 11.8 | 12.2 | - | - | 11.9 | 11.7 | 11.9 | 11.9 | - |

−45 | - | - | - | 8.9 | - | 8.8 | 8.7 | - | - | - | 8.8 | 8.8 |

−50 | - | - | 8.8 | 5.9 | 5.3 | - | - | 5.7 | 5.1 | 5.4 | 5.8 | - |

−55 | - | - | - | 3.2 | - | 2.1 | 2.1 | - | - | - | 1.3 | 1.3 |

−60 | - | - | 1.8 | 0.2 | - | - | - | - | −0.8 | −3.2 | −3.4 | - |

−65 | - | - | - | - | - | - | - | - | - | - | −8.2 | −10.9 |

−70 | - | - | - | −4.9 | - | - | - | - | - | −12 | −13.6 | - |

−75 | - | - | - | - | - | - | - | - | - | - | −20.2 | −29.4 |

−80 | - | - | - | −8.4 | - | - | - | - | - | −20.6 | −27 | - |

−85 | - | - | - | - | - | - | - | - | - | - | −31.4 | −47.8 |

−90 S | - | - | - | −9.3 | - | - | - | - | - | −25 | −33.1 | - |

*) Re-corrected to mean terrain height above sea level.

[

Obviously, the results of well-known climate researchers for the average temperature of the near- surface air temperature published during the second half of the 19^{th} century and the first two decades of the 20^{th} century are notably higher than those derived from the HadCRUT4, Berkeley, and NASA GISS

records. The goal our paper is, therefore, to assess the results of von Hann and his fellow climate researchers. Since Defant and Obst [

Since three different temperature scales related to de Réaumur (e.g., Dove), Celsius (e.g., von Hann), and Fahrenheit (e.g., Buchan) were used, we used the original sources to check all datasets adopted and converted by others. No serious conversion error was detected.

The average over the Earth’ surface reads [42 - 44]

〈 ψ 〉 = ∫ Ω ψ ( r , θ , φ ) r 2 ( θ , φ ) d Ω ∫ Ω r 2 ( θ , φ ) d Ω = ∫ 0 2 π ∫ 0 π ψ ( r , θ , φ ) r 2 ( θ , φ ) sin θ d θ d φ ∫ 0 2 π ∫ 0 π r 2 ( θ , φ ) sin θ d θ d φ (1)

Here, ψ ( r , θ , φ ) is an arbitrary variable, r ( θ , φ ) is the radius, Ω = 4 π is the solid angle of the entire planet, and d Ω = sin θ d θ d φ is the differential solid angle, where θ and φ are the zenith and azimuthal angles, respectively, in a spherical coordinate frame (

For a spherical shape of the Earth, we have r E = r = c o n s t . and, hence, ψ ( r , θ , φ ) = ψ ( θ , φ ) . Usually, r E is represented by the volumetric mean radius of r E , V ≅ 6371.0 km . Thus, Equation (1) can be written as [10,42,44,46]

〈 ψ 〉 = r E 2 ∫ 0 2 π ∫ 0 π ψ ( θ , φ ) sin θ d θ d φ r E 2 ∫ 0 2 π ∫ 0 π sin θ d θ d φ = 1 4 π ∫ 0 2 π ∫ 0 π ψ ( θ , φ ) sin θ d θ d φ (2)

Since the average along a parallel of latitude, i.e., the zonal average, is defined by [44,46 - 49]

ψ ¯ ( θ ) = 1 2 π ∫ 0 2 π ψ ( θ , φ ) d φ (3)

Equation (2) can be written as

〈 ψ 〉 = 1 4 π ∫ 0 2 π ∫ 0 π ψ ( θ , φ ) sin θ d θ d φ = 2 π 4 π ∫ 0 π sin θ d θ ( 1 2 π ∫ 0 2 π ψ ( θ , φ ) d φ ) = 1 2 ∫ 0 π ψ ¯ ( θ ) sin θ d θ (4)

Inserting the averages for both the northern and the southern hemispheres given by [

〈 ψ 〉 N H = 1 2 π ∫ 0 2 π ∫ 0 π / 2 ψ ( θ , φ ) sin θ d θ d φ = ∫ 0 π / 2 ψ ¯ ( θ ) sin θ d θ (5)

and

〈 ψ 〉 S H = 1 2 π ∫ 0 2 π ∫ π / 2 π ψ ( θ , φ ) sin θ d θ d φ = ∫ π / 2 π ψ ¯ ( θ ) sin θ d θ (6)

leads to

〈 ψ 〉 = 1 2 ( 〈 ψ 〉 N H + 〈 ψ 〉 S H ) (7)

When we set, for instance, ψ ( θ , φ ) = T ( θ , φ ) ,we obtain the globally averaged near-surface air temperature

〈 T 〉 = 1 2 ∫ 0 π T ¯ ( θ ) sin θ d θ = 1 2 ( 〈 T 〉 N H + 〈 T 〉 S H ) (8)

Since the angle of the latitude, ϕ ,and the zenith angle, θ ,are related to each other by ϕ = π / 2 − θ ,Equation (8) can be written as

〈 T 〉 = 1 2 ∫ − π / 2 π / 2 T ¯ ( ϕ ) cos ϕ d ϕ = 1 2 ( 〈 T 〉 N H + 〈 T 〉 S H ) (9)

with

〈 T 〉 N H = ∫ 0 π / 2 T ¯ ( ϕ ) cos ϕ d ϕ (10)

and

〈 T 〉 S H = ∫ − π / 2 0 T ¯ ( ϕ ) cos ϕ d ϕ (11)

The Earth is, of course, not a sphere, but in a first approximation an oblate ellipsoid of revolution, i.e. an oblate spheroid, flattened at the poles and bulging at the Equator due to centrifugal forces. This oblate spheroid serves as a surface of reference for the mathematical reduction of geodetic and cartographic data [

x 2 a 2 + z 2 c 2 = 1 (12)

The coordinates are given by x = a cos ( π 2 − θ ) = a sin θ and z = c sin ( π 2 − θ ) = c cos θ . The ellipsoidal radius is

r ( θ ) = r E , V + Δ r ( θ ) = r E , V ( 1 + Δ r ( θ ) r E , V ) (13)

where Δ r ( θ ) is the difference between r ( θ ) and r E , V of the Earth. The ratio δ = Δ r ( θ ) / r E , V ranges from δ = − 0.00223 at the Poles to δ = 0 .00111 at the Equator (

A E = ∫ 0 2 π ∫ 0 π r 2 ( θ , φ ) sin θ d θ d φ = 2 π r E , V 2 ∫ 0 π ( 1 + δ ) 2 sin θ d θ (14)

Since the surface of an oblate spheroid is also given by [

A E = 2 π a ( a + c 2 a 2 − c 2 arsinh ( a 2 − c 2 c ) ) (15)

or in the re-arranged form

A E = 2 π a 2 + π c 2 e ln ( 1 + e 1 − e ) (16)

with e = 1 − ( c / a ) 2 ,we have

∫ 0 π ( 1 + δ ) 2 sin θ d θ = 2 (17)

The equivalent spherical radius is r E , A ≅ r E , V ≅ 6371.0 km . In the case of a sphere, the rule A 3 = 36 π V 2 is exactly fulfilled, where A and V are the surface and the volume, respectively. Since of all surfaces of the same area, the sphere has the greatest volume, we have A 3 ≥ 36 π V 2 [

〈 ψ 〉 = 1 4 π ∫ 0 2 π ∫ 0 π ψ ( δ , θ , φ ) ( 1 + δ ) 2 sin θ d θ d φ (18)

Using again the zonal average defined by Equation (3) leads to

〈 ψ 〉 = 1 2 ∫ 0 π ψ ¯ ( θ ) ( 1 + δ ) 2 sin θ d θ (19)

Inserting the averages for both the Northern and Southern Hemispheres defined by

〈 ψ 〉 N H = ∫ 0 π / 2 ψ ¯ ( θ ) ( 1 + δ ) 2 sin θ d θ (20)

and

〈 ψ 〉 S H = ∫ π / 2 π ψ ¯ ( θ ) ( 1 + δ ) 2 sin θ d θ (21)

leads to

〈 ψ 〉 = 1 2 ( 〈 ψ 〉 N H + 〈 ψ 〉 S H ) (22)

Thus, the globally averaged near-surface air temperature is given by

〈 T 〉 = 1 2 ∫ 0 π T ¯ ( θ ) ( 1 + δ ) 2 sin θ d θ = 1 2 ( 〈 T 〉 N H + 〈 T 〉 S H ) (23)

As in the case of spherical averaging, θ can be substituted by ϕ .

Hansen and Lebedeff [

“The principal limitation of this data set for global or hemispheric analysis is the incomplete spatial coverage, illustrated in

However, as expressed by Equations (8) and (9), the uncertainty regarding observations in the south-polar region is of minor importance in determining the globally averaged near-surface temperature. Furthermore, von Hann and his fellow climate researchers not only gathered data from meteorological stations, but also eyewitness descriptions of weather and climate. Over land, the near-surface dry- and wet-bulb temperatures as well as the maximum and minimum temperatures were measured in thermometer boxes (Stevenson’s screens) at a height of 2 m or so above the ground using a standardized thermometer equipment as illustrated in

Measurements of the sea-surface temperature have been performed since the mid-18^{th} century; mainly wooden and canvas buckets served to take water probes. Krümmel [

Nonetheless, even today, network density and design affect the regional averages over landscapes. In 2009, PaiMazumber and Mölders [^{−1} (0.5 m∙s^{−1}), and 0.2 mm day^{−1} and overestimates regional averages of 2 m temperature, downward shortwave radiation, and soil temperature up to 1.9 K (1.4 K), 19 W∙m^{−2} (14 W∙m^{−2}), and 1.5 K (1.8 K) in July (December) [

Alexander von Humboldt was the first who drew annual isotherms on a map of the Earth in 1817 [

“Humboldt withheld publishing his idea in the form of a generalized world map while he waited for data from weather stations around the world. Thus, the isotherm concept remained virtually unknown outside of the broader scientific community until 1838, when it was given wider recognition on the map below [here

Batchelder [

“Durch den Umstand, dass ein Parallelkreis teils über Land, teils über Wasser verläuft, entstehen klimatische Unterschiede zwischen West und Ost, oder Verschiedenheiten des Klimas nach den Meridianen, welche im solaren Klima nicht vorhanden wären. Neben der ungleichen Erwärmung und Erkaltung von Wasser und Land werden ausserdem durch das Vorhandensein des Landes gewisse konstante Luft- und Meeresströmungen erzeugt, welche ebenfalls eine Verschiedenheit des Klimas unter verschiedenen Meridianen desselben Parallels bedingen.”

The translation from German into English given by Batchelder [

“Owing to the fact that a circle of latitude passes partly over land and partly over water, there arise climatic differences between West and East, or variations of the climate according to the longitudes, that in the solar climate would not exist. Following the dissimilar warming and cooling of water and land there are, moreover, set up, owing to the presence of the land, certain constant air and sea currents which also make possible a difference of climate on different meridians of the same parallel.”

The notion “solar climate” is discussed in Subsection 3.3.

Von Hann [

T ( ϕ ) = A + B cos m ( ϕ ) ︸ waterglobe + C n cos ( 2 ϕ ) ︸ landeffect (24)

where n is the fraction of land compared to the circumference of the respective parallel. The coefficients A, B, C, and the exponent m must be derived from observations. Forbes [

“Between latitude 40˚ and latitude 50˚ is the parallel on which both water and land hemispheres have the same temperature. In higher latitudes, the water hemisphere is warmer than the land hemisphere. Even Dove’s isothermal charts of the northern hemisphere show that the transition takes place between latitudes 40˚ and 45˚, and the graphic representation, above referred to, shows more precisely that this transition occurs at latitude 42.5˚. Beyond this latitude, as far as the pole, a water hemisphere is warmer than a hemisphere wholly covered by land.”

In his paper on von Hann’s contribution to modern climatology, Kahlig [

Isothermal charts as developed by Dove [

[

DeCourthy Ward [

More than 85 years later, Levitus [

Spitaler [

“Die Grundlage für die vorliegende Untersuchung lieferten die neuen Isothermenkarten von Wild und Prof. Hann, welche nach dem sämtlichen bis jetzt vorliegenden Beobachtungsmaterial der Erde gezeichnet wurden. Ich habe für jeden 10. Breitengrad von 5 zu 5 Längengraden, für die dazwischenliegenden Breitengrade aber nur für jeden 10. Längengrad die Temperatur des Jahresmittels bestimmt, sowie die mittleren Temperaturen der beiden extremen Monate Januar and Juli graphisch interpoliert und auf diese Weise einerseits aus je 72, andererseits aus je 36 äquidistanten Temperaturwerten die normale Temperatur der Breitenkreise bestimmt.”

The translation reads:

“The basis for the present study was the new isothermal charts of Wild and Prof. Hann, which were drawn based on all the Earth’s observation material available so far. For each 10th degree of latitude and for 5 to 5 degrees of longitude, but for the intermediate latitudes only for each 10th degree of longitude, I determined the annual-average temperature and interpolated the mean temperatures of the two extreme months of January and July graphically. In doing so, I determined the normal temperature of the parallels of latitude based on 72 and 36 equidistant temperature values, respectively.”

Batchelder [^{th} degree of latitude, from 80˚N to 50˚S, by averaging the observed temperatures at 36 equidistant points or stations on the circle, i.e., at every 10^{th}meridian. The temperatures at these 36 points were determined, when an isotherm did not happen to fall directly over the point, by interpolation between the nearest isotherms. Referring to Forbes [

“Although one authority considers that ‘It is by no means an easy matter to deduce the mean temperature of a given parallel correctly from an isothermal chart,’ yet as Buchan constructed his isotherms at intervals of 5˚F, interpolation was usually simple and direct, and the limit of error rarely rose above one degree. The most unsatisfactory cases of interpolation were those succession of isotherms) as in July, 130˚W, 50˚N. Still, even here, 1˚F seems usually the error. A peculiar case occurs in January over the northern Amazon basin; as the observer moves north ward over South America he finds the temperature fall from the central part until above the equator, where occurs an isolated area of +85˚ over the Isthmus of Panama. The neighborhood of this area is very difficult to interpolate for; to the north the nearest isotherm is that of 75˚, a ten-degree skip, to the east and west are no neighboring isotherms, and far away to the south is the isotherm of 80˚ on a falling gradient. In the case of the parallel of 80˚N on the Annual Chart, too, the limit of error between 100˚W and 130˚W may rise as high as 4˚F. On the equator, owing to the slight gradient, the general limit of error is about 2˚F.”

Hopfner [^{th} degree of latitude, but from 80˚N to 60˚S by averaging the observed temperatures at 36 equidistant points or stations on the respective parallel of latitude. He also discussed the accuracy of the use of isothermal charts in detail.

As argued by Köppen [

As listed in

The reduction of temperatures to sea level seems to be a problem. Even though the reduction of air pressure to sea level is widely accepted, this is not the case of near-surface air temperature. In his textbook “Lehrbuch der Meteorologie“ published in 1906, von Hann [

Latitude in ˚ | |||||||||
---|---|---|---|---|---|---|---|---|---|

80 | 60 | 40 | 20 | 0 | −20 | −40 | −60 | −80 | |

Maximum | −10 | 7 | 17 | 29 | 28 | 25 | 14 | 2 | −14 |

Minimum | −20 | −8 | 9 | 23 | 24 | 18 | 9 | −5 | −24 |

Difference | 10 | 15 | 8 | 6 | 4 | 7 | 5 | 7 | 10 |

“Die Isothermen stellen die Temperaturverteilung auf der Erde so dar, wie wenn alle Orte, nach deren Temperaturaufzeichnungen sie entworfen worden sind, im Meeresniveau liegen würden. Die Temperaturmittel müssen daher, bevor man sie auf der Karte einträgt, auf das Meeresniveau reduziert werden, wozu die früher angeführten Erfahrungen über die Wärmeabnahme mit der Höhe benützt werden.”

The translation reads:

“The isotherms represent the temperature distribution on Earth as if all the locations, which contributed to them with their temperature records, were at sea level. The temperature averages must, therefore, be reduced to sea level before entering the chart to which the previously mentioned experience about the heat decrease with height are used.”

Buchan [

“The mean temperature falls about 5.5˚C for every 1000 metres of ascent (1˚F per 330 feet or 16˚F per mile); but, in the Northern Hemisphere, between 30˚N and 70˚N, the mean temperature decreases only 0.75˚C (1.35˚F) for every increase of a degree of latitude; i.e. 0.0068˚C for every kilometre nearer the pole (0.0197˚F per mile). The temperature therefore falls 800 times more in the same distance in a vertical direction than in a horizontal one. This may be expressed by saying that the vertical temperature gradient is 800 times steeper than the horizontal one. This contrast can be seen in the sections on Plate 1, where the vertical distances are much exaggerated when compared with the horizontal ones, and hence the sections greatly minimise the contrast.”

Note that von Hann [

Sellers [

“Temperatures given in [his]

We considered Arrhenius’ [14,26] data re-corrected to mean terrain height above sea level and Sellers’ [

Sellers considered the physical and climatological data of 10˚-latitude zones. Thus, we assigned his temperature data to the center of the respective latitude zones.

The normal temperatures determined by the authors considered in this study are not completely independent. As aforementioned, subsequent authors improved them using the rapidly increased number of weather observations in the 2^{nd} half of the 19^{th} century (

Graphic solutions have a long tradition in science and technology. Cremona-Maxwell diagrams, for instance, were used in statics with sufficient accuracy long before Konrad Zuse developed the first programmable computer in the late 1930s.

Forbes’ [

“If the surface of the earth were occupied altogether by land, and if there were no surrounding atmosphere, the condition of our planet would be somewhat similar to that of the moon at the present time. Under these conditions, the distribution of temperature over the earth would depend solely upon the amount of heat received from the sun at any given place, and upon the loss of heat by radiation at that place. As these two factors would necessarily be the same at all points along the same parallel of latitude, the zones of equal temperature would coincide with the parallels of latitude. Even the presence of a vapourless atmosphere would interfere but little with this

distribution of temperature, for only the absolute amounts of heat received at, and radiated from, the surface of the earth would thereby be affected. It is true that convectional currents would be produced under these conditions; but as there would be no reason for the more frequent occurrence of warm or cold air currents along some meridians than others, the distribution of temperature in zones bounded by the parallels of latitude would not thereby be interfered with.”

The solar climate depends on the astronomic aspects like the distance between the Sun’s center and the Earth (Earth-Moon barycenter), the obliquity of Earth’s rotation axis with respect to the normal of the ecliptic plane, the angular velocity of the rotation, and the total solar irradiance (TSI) for 1 AU. Only the absorbed solar radiation and the thermal effect related to the regolith should be considered. With respect to geological time scales, also changes in the precession of Earth’s rotation axis, and long-term variations of the eccentricity, obliquity, and precession of the Perihelion caused by the Sun, Moon and planets of our solar system must be considered [69 - 75].

ϑ d d t 〈 c ρ T s l a b 〉 V = ( 1 − α ( Θ 0 , θ , φ ) ) F cos Θ 0 − ε ( θ , φ ) σ T s l a b 4 ( θ , φ ) − H s l ( θ , φ ) ,(25)

where 〈 … 〉 V is the volume average.Here, t is time, T s l a b , ρ ,and c, are the temperature, bulk density, and specific heat of this slab of soil, respectively. Furthermore, ( 1 − α ( Θ 0 , θ , φ ) ) F cos Θ 0 is the solar radiation absorbed by the slab, where F is the solar irradiance reaching the surface, α ( Θ 0 , θ , φ ) is the integral albedo inthe solar range, and Θ 0 is the local zenith distance of the Sun’s center. Kramm et al. [

α ( Θ 0 ) = α 0 + ( Θ 0 45 ∘ ) 3 ( a + b ( Θ 0 45 ∘ ) 5 ) (26)

where α 0 = 0.10 is the normal albedo, and a = 0.045 and b = 5.47 × 10 − 4 are empirical values. With exception of Keihm’s value for b, all other values are based on observations from the Lunar Reconnaissance Orbiter Diviner Lunar Radiometer Experiment [77,78]. In the case of the Earth, cos Θ 0 was determined using the rules of spherical trigonometry (e.g., [74,75,79,80])

cos Θ 0 = sin ϕ sin δ S u n + cos ϕ cos δ S u n cos h = cos θ sin δ S u n + sin θ cos δ S u n cos h (27)

Here, δ S u n is the declination of the Sun, ϕ is latitude, and h is the hour angle from the local meridian. The solar irradiance, F, is given by

F = ( r S u n r ) 2 F S u n (28)

where F S u n is the solar emittance [74,75,81,82], r S u n ≅ 6.963 × 10 5 km [

S = ( r S u n r 0 ) 2 F S u n (29)

Combining formulae (28) and (29) yields

F = ( r 0 r ) 2 S (30)

Furthermore, F I R = ε ( θ , φ ) σ T s l a b 4 ( θ , φ ) is the emitted radiation according to the power law of Stefan

[^{−}^{2} at 1 AU was taken from Kopp and Lean [^{−2} during 1850 to 2015. Thus, the solar climate derived by Kramm et al. [

formula (Equation (24)) would provide 〈 T s e a 〉 = 13.7 ˚ C for an aqueous globe and 〈 T l a n d 〉 = 20.7 ˚ C for a

rocky one.

The slab-temperature distribution predicted by Kramm et al. [

global average of this slab-temperature distribution is 〈 T s l a b 〉 = − 52.4 ˚ C = 220.7 K . Since 〈 T s l a b 4 ( θ , φ ) 〉 1 / 4 = 266.4 K ,one obtains 〈 T s l a b 〉 = 0.828 〈 T s l a b 4 ( θ , φ ) 〉 1 / 4 for the obliquely rotating Earth in the absence of its atmosphere.

Q ¯ ( θ ) from θ = 0 to θ = π yields, of course, 〈 Q 〉 = 279 .7 W ⋅ m − 2 ,mentioned before.

Defant and Obst [

and T F ( θ ) are illustrated in

This procedure, however, is incorrect from physical and mathematical points of view. Since the local radiation balance is given by [

Q ( θ , φ ) = ( 1 − α ( Θ 0 , θ , φ ) ) F cos Θ 0 = σ T s 4 ( θ , φ ) (31)

where Q ( θ , φ ) is the absorbed solar irradiance, and T s is the local surface temperature, zonal averaging would provide (see Equation (3))

Q ¯ ( θ ) = 1 2 π ∫ 0 2 π ( 1 − α ( Θ 0 , θ , φ ) ) F cos Θ 0 d φ = σ 2 π ∫ 0 2 π T s 4 ( θ , φ ) d φ (32)

and, hence,

Q ¯ ( θ ) = σ T s 4 ¯ ( θ ) (33)

In accord with the general inequality of Gerlich and Tscheuschner [

T s ¯ = ∫ X T s d W ≤ ∫ X T s 4 d W 4 = T s 4 ¯ 4 (34)

for a non-negative measurable function T s and a probability measure W, the zonal average of the surface temperature, fulfills the inequality

T s ¯ ( θ ) ≤ T Q ( θ ) = Q ¯ ( θ ) / σ 4 = T s 4 ¯ ( θ ) 4 (35)

In the case of completely absorbed solar irradiance, i.e., α ( Θ 0 , θ , φ ) = 0 ,we would have Q ¯ ( θ ) = F ¯ ( θ ) and, hence, T Q ( θ ) = T F ( θ ) . Thus, as illustrated in

radiation temperature T F ( θ ) . Based on Equation (4), the integration of T s ¯ ( θ ) from θ = 0 to θ = π

provided 〈 T s 〉 = 157.2 K . This global mean surface temperature substantially agrees with result of Gerlich and Tscheuschner [

As illustrated by

by Gerlich and Tscheuschner [

Obviously, the existence of the atmosphere, the nonuniformity of the Earth’s surface, the uneven ocean-land distribution, the airflows and ocean currents cause the transition from the solar climate to the real climate [

The atmosphere is a complex thermo-fluid dynamic system with various degrees of freedom. It consists of various layers (e.g., troposphere, stratosphere, mesosphere, thermosphere) separated by conceptual partitions called pauses [^{18} kg is below the altitude of 30 km above sea level. The atmospheric response time to an imposed change is of the order of days or weeks owing to its notable compressibility and its low density and specific heat [^{24} J, i.e., less than 0.1% of the total energy of the oceans.

Based on recent observations, the solar insolation at the top of the atmosphere (TOA) corresponds to a globally averaged value of about 〈 F 〉 T O A = 340.3 W ⋅ m − 2 (100 units). As sketched in

As sketched in ^{−}^{2}) and clouds (4 units or 13.6 W∙m^{−2}). Especially the absorption of solar radiation by molecular oxygen (O_{2}) and ozone (O_{3}) heats the atmosphere directly [75,94]. Water vapor (H_{2}O) and O_{2} are also active in the visible and near infrared range; nitrogen dioxide (NO_{2}) is active in the visible range, too. Furthermore, a considerable portion of the solar radiation is back-scattered by molecules (Rayleigh scattering), cloud and aerosol particles (Lorenz-Mie scattering). A notable amount of solar radiation reaching the Earth’s surface is reflected, either by the soil-vegetation and water systems on land or by the oceans. These processes contribute to a planetary albedo of about 30 units (102.1 W∙m^{−2}), on global average (^{−2}) of solar radiation, on global average, feed the EAS with energy. However, only about 46 units (156.5 W∙m^{−2}) are absorbed by water (including ice) and land masses (including vegetation) in the close vicinity of the Earth’s surface (

Atmospheric motions are stochastic to a certain extent. However, organized patterns like Rossby waves, mountain-induced gravity and inertial-gravity waves, cyclones and anticyclones, jet streams, and circulation pattern of different sizes like Hadley, Ferrel, and polar cells, monsoonal circulation, small-scale land-sea breezes, convection roles, etc. can be identified. Variations of these flow patterns may affect the energy conversion at the Earth’s surface on local and regional scales. Turbulent motion can mainly be observed in the atmospheric boundary layer (ABL) and along jet streams. The exchange of sensible and latent heat between the land or water masses adjacent to the Earth’s surface and the atmosphere is strongly controlled by molecular and turbulent transfer processes within the Prandtl layer (also called the atmospheric surface layer, ASL). Over canopies of tall vegetation, the outer edge of the Prandtl layer may be at 100 m height above ground. The climates of landscapes may be described by the Köppen-Geiger climate classification. An updated version was derived by Peel et al. [

During cloud formation huge amounts of latent heat are released that heat the ambient air directly. Clouds of various horizontal and vertical extensions and compositions of hydrometeors mainly occur in the troposphere. They strongly interact with both solar and infrared radiation by absorption, scattering, and emission (only infrared radiation). Clouds also affect the energy conversion at the interface Earth- atmosphere via radiation and by precipitation thereby altering the surface properties of vegetation and soils. Precipitation contributes in small amounts to the conversion of potential energy into kinetic energy which is finally converted into heat.

Solar radiation absorbed by water and soil layers adjacent to the Earth’s surface is converted into heat. Hence, it contributes to warming these layers [^{−}^{2}) and latent heat (23 units or 78.3 W∙m^{−}^{2}). These fluxes, on global average, heat the atmosphere from below and cause convective transports of energy and mass into the upper troposphere. There, especially the release of latent heat during phase transition processes contributes to establish atmospheric circulation systems of different spatial and temporal scales [75,93].

As the absorption of solar radiation by atmospheric constituents and the exchange of energy between the soil and/or water layers at the Earth-atmosphere interface by the fluxes of sensible and latent heat already heated the atmosphere (about 177.0 W∙m^{−}^{2} of the energetically relevant solar radiation, on global average), we have to expect that gaseous atmospheric constituents able to emit and absorb infrared (IR) radiation in finite spectral ranges, will emit energy in the IR range in all directions. The amount of this IR radiation depends on the local temperature of the mixture of these constituents. Therefore, it is indispensable to consider the down-welling IR radiation reaching the Earth’s surface, where most of it is absorbed. The same is true for hydrometeors.

The water and soil layers adjacent to the Earth-atmosphere interface, of course, also emit IR radiation depending on their local temperatures. The net emission in the IR range (emitted radiation minus absorbed down-welling radiation) is about 18 units (61.3 W∙m^{−}^{2}), on global average. A notable portion of this IR net emission is absorbed by atmospheric constituents and emitted in all directions, too. A small fraction propagates through the atmosphere (about 6 units or 20.4 W∙m^{−}^{2}) with marginal extinction by intervening constituents. Such a spectral region is the so-called atmospheric window ranging from 8.3 μm to 12.5 μm (e.g., [74,75,82,97,98]). It only contains the 9.6 μm-band of ozone. Satellite-borne radiometers use the atmospheric window between 10 μm and 12.5 μm to measure radiation up-welling from the Earth’s surface [

Gases like H_{2}O, carbon dioxide (CO_{2}), and O_{3} and hydrometeors also emit IR radiation to space. As shown in ^{−}^{2}), and the emission by clouds to space is 26 units (88.5 W∙m^{−}^{2}).

Besides the solar climate, the oceanic circulation affects the meridional distribution of normal temperatures [

The role of the thermohaline circulation is assessed somewhat controversy. Wunsch [

To calculate the global and the hemispheric averages of the near-surface air temperature, we used Equation (8) and the datasets of the historical climatological and annual averages of temperature along numerous parallels of latitude(i.e., the zonal averages of temperature or normal temperatures) published between 1852 (Dove [

I ( θ ) = I 0 + A 1 + exp ( − θ − θ c + B / 2 C ) ( 1 − 1 1 + exp ( − θ − θ c − B / 2 D ) ) (36)

because the temperature distributions illustrated by

Author(s), and dataset | Average near-surface air temperature in ˚C | |||||||
---|---|---|---|---|---|---|---|---|

Polygons | Equation (36) | |||||||

NH | SH | Δ(NH-SH) | Earth | NH | SH | Δ(NH-SH) | Earth | |

Dove [ 19 ], Do1852 | 14.9 | 11.8 | 3.1 | 13.3 | 15.3 | 14.3 | 1.0 | 14.8 |

Forbes [ 23 ], Fo1859 | 14.9 | 11.8 | 3.1 | 13.4 | 15.4 | 14.1 | 1.3 | 14.8 |

Ferrel [ 12 ], Fe1877 | 14.8 | 15.4 | −0.6 | 15.1 | 15.2 | 15.9 | −0.7 | 15.5 |

Spitaler [ 10 ], Sp1885 “Complete dataset” | 15.2 | 14.6 | 0.6 | 14.9 | 15.2 | 14.7 | 0.5 | 14.9 |

Spitaler [ 10 ], Sp1885 “Reduced dataset” | 14.7 | 14.1 | 0.6 | 14.4 | 15.2 | 14.7 | 0.5 | 15.0 |

Batchelder [ 11 ], Ba1894 | 14.7 | 13.3 | 1.4 | 14.0 | 15.2 | 14.0 | 1.2 | 14.6 |

Arrhenius [ 26 ], A1896-1 | 15.0 | 13.5 | 1.5 | 14.2 | 15.2 | 13.7 | 1.5 | 14.4 |

Arrhenius [ 14 , 26 ]*, A1896-2 | 13.7 | 13.0 | 0.7 | 13.3 | 13.8 | 13.2 | 0.6 | 13.5 |

von Bezold [ 7 ], vB1901-1 “Spitaler” | 15.3 | 14.4 | 0.9 | 14.9 | 15.0 | 14.1 | 0.9 | 14.6 |

von Bezold [ 7 ], vB1901-2 “Batchelder” | 15.4 | 14.4 | 1.0 | 14.9 | 15.1 | 14.0 | 1.1 | 14.6 |

Hopfner [ 33 ], Ho1906-1 | 14.9 | 13.2 | 1.7 | 14.1 | 15.4 | 13.6 | 1.8 | 14.5 |

Hopfner [ 33 ] Ho1906-2 | 15.0 | 13.7 | 1.3 | 14.4 | 15.5 | 14.1 | 1.4 | 14.8 |

von Hann [ 32 ], vH1908 | 14.8 | 13.6 | 1.2 | 14.2 | 15.2 | 14.2 | 1.0 | 14.7 |

von Bezold [ 29 ], vB1906, Börnstein [ 35 ], Bö1913, “Complete dataset” | 15.5 | 13.7 | 1.8 | 14.5 | 15.1 | 13.6 | 1.5 | 14.3 |
---|---|---|---|---|---|---|---|---|

von Bezold [ 29 ], vB1906, Börnstein [ 35 ] Bö1913, “Reduced dataset” | 15.4 | 13.5 | 1.9 | 14.5 | 15.1 | 13.5 | 1.6 | 14.3 |

Defant and Obst [ 36 ], De1923 | 14.7 | 13.1 | 1.6 | 13.9 | 15.2 | 13.7 | 1.5 | 14.5 |

Köppen [ 37 ], T_{min} | 9.1 | 7.8 | 1.3 | 8.4 | 11.4 | 10.5 | 0.9 | 10.9 |

Köppen [ 37 ], T_{max} | 17.0 | 13.8 | 3.2 | 15.4 | 19.6 | 16.4 | 3.2 | 18.0 |

von Hann-Süring [ 38 ], vHS1939 | 15.1 | 13.2 | 1.9 | 14.2 | 14.9 | 13.2 | 1.7 | 14.0 |

Haurwitz and Austin [ 40 ] | 14.7 | 12.5 | 2.2 | 13.6 | 15.1 | 13.1 | 2.0 | 14.1 |

Sellers [ 39 ], Se1965 | 12.7 | 11.1 | 1.6 | 11.9 | 13.1 | 11.8 | 1.3 | 12.5 |

Kramm et al. [ 44 ] | −52.0 | −52.8 | 0.8 | −52.4 | −51.9 | −52.7 | 0.8 | −52.3 |

*)Re-corrected to mean terrain height above sea level.

Author(s), and dataset | Number of points | I_{0} | θ_{c} | A | B | C | D |
---|---|---|---|---|---|---|---|

Dove [ 19 ], Do1852 | 14 | −320.57 | 1.5636 | 1438.28 | 1.1416 | 0.8725 | 0.8819 |

Forbes [ 23 ], Fo1859 | 14 | −270.56 | 1.5671 | 957.38 | 1.8252 | 0.7443 | 0.7501 |

Ferrel [ 12 ], Fe1877 | 15 | −310.94 | 1.5710 | 1186.28 | 1.5531 | 0.8301 | 0.8321 |

Spitaler [ 10 ], Sp1885 “Complete dataset” | 33 | −327.05 | 1.5543 | 1454.92 | 1.1475 | 0.8728 | 0.8993 |

Spitaler [ 10 ], Sp1885 “Reduced dataset” | 19 | −340.78 | 1.5502 | 1764.13 | 0.7686 | 0.9096 | 0.9359 |

Batchelder [ 11 ], Ba1895 | 14 | −317.30 | 1.5606 | 1429.10 | 1.1394 | 0.8664 | 0.8802 |

Arrhenius [ 26 ], A1896-1 | 13 | −264.31 | 1.5653 | 918.09 | 1.8838 | 0.7242 | 0.7367 | |
---|---|---|---|---|---|---|---|---|

Arrhenius [ 14 , 26 ]*, A1896-2 | 13 | −278.01 | 1.5673 | 991.41 | 1.7855 | 0.7595 | 0.7681 | |

von Bezold [ 7 ], vB1901-1, “Spitaler” | 38 | −303.90 | 1.5608 | 1150.66 | 1.5838 | 0.8090 | 0.8300 | |

von Bezold [ 7 ], vB1901-2, “Batchelder” | 38 | −275.19 | 1.5652 | 986.81 | 1.7810 | 0.7526 | 0.7656 | |

Hopfner [ 33 ], Ho1906-1 | 14 | −259.62 | 1.5653 | 901.82 | 1.8959 | 0.7169 | 0.7277 | |

Hopfner [ 33 ], Ho1906-2 | 15 | −323.18 | 1.5568 | 1432.82 | 1.1634 | 0.8690 | 0.8891 | |

von Hann [ 32 ], vH1908 | 19 | −298.00 | 1.5620 | 1151.37 | 1.5485 | 0.8076 | 0.8246 | |

von Bezold [ 29 ], vB1906 Börnstein [ 35 ], Bö1913 “Complete dataset” | 41 | −240.11 | 1.5664 | 809.04 | 2.0198 | 0.6693 | 0.6810 | |

von Bezold [ 29 ], Börnstein [ 35 ], “Reduced dataset” | 21 | −254.79 | 1.5658 | 868.60 | 1.9525 | 0.7012 | 0.7127 | |

Defant and Obst [ 36 ], De1923 | 19 | −267.45 | 1.5656 | 942.99 | 1.8404 | 0.7356 | 0.7463 | |

Köppen [ 37 ], T_{min} | 9 | −311.71 | 1.5586 | 1605.24 | 0.8397 | 0.8809 | 0.8958 | |

Köppen [ 37 ], T_{max} | 9 | −357.88 | 1.5420 | 1691.43 | 0.9473 | 0.9122 | 0.9469 | |

von Hann-Süring [ 38 ], vHS1939 | 37 | −271.36 | 1.5658 | 957.96 | 1.8258 | 0.7435 | 0.7516 | |

Haurwitz and Austin [ 40 ] | 19 | −313.92 | 1.5588 | 1341.92 | 1.2678 | 0.8513 | 0.8675 | |

Sellers [ 39 ], Se1965 | 18 | −242.72 | 1.5693 | 831.23 | 1.9678 | 0.6893 | 0.6859 | |

Kramm et al. [ 44 ] | 37 | −77.91 | 1.5691 | 376.50 | 2.0028 | 0.4250 | 0.4255 |

*) Re-corrected to mean terrain height above sea level.

With respect to the polygons, Köppen’s [

Obviously, the poor coverage of the Southern Hemisphere by observations during that time indicated by the normal temperatures of Dove [

〈 T 〉 S H = 15.4 ˚ C (Fe1877 [

On the contrary, the average for the Northern Hemisphere based on Dove [

For comparison: The datasets of Defant and Obst [

〈 T 〉 N H = 15.1 ˚ C for vHS1939, respectively. Sellers’ [

As illustrated in ^{th} degree of latitude, while Blüthgen [

By using Equation (36), we fitted the meridional distributions of the points of the integrand T ¯ ( θ ) sin θ related to the historical distributions of normal temperatures. The parameters of our fitting procedure are listed in

For comparison: Based on Equation (36), the datasets of Defant and Obst [

To estimate the uncertainty of our results, the numerical solution was repeated 50 times to create an ensemble of 50 realizations, where for each solution the zonal averages of the temperature for all parallels of latitude were randomly modified by adding temperature values, Δ R T ,that are normally distributed within a standard deviation of σ = ± 2 K . No seed was presupposed. This procedure was applied to each of the datasets of the zonal averages of temperature.

By using Equation (8), the integration of the polygon provided, for instance, for vB1906 and Bö1913 〈 T 〉 = 14.5 ˚ C , 〈 T 〉 S H = 13.7 ˚ C , 〈 T 〉 N H = 15.5 ˚ C ,and Δ ( N H − S H ) = 1.8 ˚ C (

lysis yielded for these datasets 〈 T 〉 = ( 14.5 ± 0.3 ) ˚ C , 〈 T 〉 S H = ( 13.6 ± 0.4 ) ˚ C , 〈 T 〉 N H = ( 15.3 ± 0.4 ) ˚ C ,and

Δ ( N H − S H ) = ( 1.7 ± 0.6 ) ˚ C (

〈 T 〉 S H = 13.2 ˚ C , 〈 T 〉 N H = 15.1 ˚ C ,and Δ ( N H − S H ) = 1.9 ˚ C (

Author(s), and dataset | Average near-surface air temperature and standard deviation in ˚C | |||
---|---|---|---|---|

Polygons | ||||

NH | SH | Δ(NH-SH) | Earth | |

Dove [ 19 ], Do1852 | 14.9 ± 0.7 | 11.7 ± 1.1 | 3.2 ± 1.3 | 13.3 ± 0.6 |

Forbes [ 23 ], Fo1859 | 15.0± 0.7 | 11.7 ± 1.1 | 3.3± 1.3 | 13.3 ± 0.6 |

Ferrel [ 12 ], Fe1877 | 14.7 ± 0.7 | 15.3 ± 0.9 | −0.6 ± 1.1 | 15.0 ± 0.6 |

Spitaler [ 10 ], Sp1885 “Complete dataset” | 15.1 ± 0.5 | 14.6 ± 0.6 | 0.4± 0.7 | 14.8± 0.4 |

Batchelder [ 11 ], Ba1894 | 14.9 ± 0.7 | 13.0 ± 0.9 | 1.9 ± 1.2 | 13.9± 0.5 |

Arrhenius [ 26 ], A1896-1 | 15.1 ± 0.8 | 13.4 ± 0.8 | 1.8± 1.1 | 14.2 ± 0.6 |

Arrhenius [ 14 , 26 ]*, A1896-2 | 13.7± 0.8 | 12.7 ± 0.8 | 1.0 ± 1.1 | 13.2 ± 0.6 |

von Bezold [ 7 ], vB1901-1 “Spitaler” | 15.3± 0.4 | 14.3 ± 0.5 | 1.0 ± 0.8 | 14.8 ± 0.3 |

von Bezold [ 7 ], vB1901-2 “Batchelder” | 15.3± 0.4 | 14.4± 0.6 | 0.9± 0.8 | 14.8 ± 0.4 |

Hopfner [ 33 ], Ho1906-1 | 15.1 ± 0.7 | 13.0 ± 0.9 | 2.1 ± 1.2 | 14.0 ± 0.5 |

Hopfner [ 33 ], Ho1906-2 | 14.9 ± 0.7 | 13.7± 0.9 | 1.2 ± 1.1 | 14.3 ± 0.5 |

von Hann [ 32 ], vH1908 | 14.7 ± 0.7 | 13.5 ± 0.8 | 1.1± 1.1 | 14.1± 0.5 |

von Bezold [ 29 ], vB1906 Börnstein [ 35 ], Bö1913 “Complete dataset” | 15.3 ± 0.4 | 13.6 ± 0.4 | 1.7± 0.6 | 14.5 ± 0.3 |

Defant and Obst [ 36 ], De1923 | 14.6 ± 0.7 | 13.1 ± 0.8 | 1.5 ± 1.1 | 13.9 ± 0.5 |

von Hann-Süring [ 38 ], vHS1939 | 15.0 ± 0.4 | 13.1 ± 0.5 | 1.9± 0.7 | 14.1 ± 0.3 |

Sellers [ 39 ], Se1965 | 12.5 ± 0.7 | 10.9± 0.7 | 1.5 ± 0.9 | 11.7 ± 0.5 |

*) Re-corrected to mean terrain height above sea level.

provided 〈 T 〉 = ( 14.1 ± 0.3 ) ˚ C , 〈 T 〉 S H = ( 13.1 ± 0.5 ) ˚ C , 〈 T 〉 N H = ( 15.0 ± 0.4 ) ˚ C ,and Δ ( N H − S H ) = ( 1.9 ± 0.7 ) ˚ C (

Presupposing an oblate spheroid, the numerical solution of Equation (23) provided for vB1906 and Bö1913 〈 T 〉 = 14.6 ˚ C , 〈 T 〉 S H = 13.7 ˚ C , 〈 T 〉 N H = 15.5 ˚ C ,and Δ ( N H − S H ) = 1.8 ˚ C . One of these values is slightly higher than that provided by Equation (8); however, the increase is mainly a rounding effect due to a marginal change in the second decimal place. Based on our uncertainty analysis, we obtained for these datasets 〈 T 〉 = ( 14.5 ± 0.3 ) ˚ C , 〈 T 〉 S H = ( 13.7 ± 0.4 ) ˚ C , 〈 T 〉 N H = ( 15.4 ± 0.4 ) ˚ C ,and Δ ( N H − S H ) = ( 1.7 ± 0.6 ) ˚ C . Again, any increase is mainly a rounding effect due to a marginal change in the second decimal place. For comparison: vHS1939 provided 〈 T 〉 = ( 14.1 ± 0.3 ) ˚ C , 〈 T 〉 S H = ( 13.2 ± 0.5 ) ˚ C , 〈 T 〉 N H = ( 15.1 ± 0.4 ) ˚ C ,and Δ ( N H − S H ) = ( 1.9 ± 0.7 ) ˚ C .

The normal temperatures of Kramm et al. [

Zonal averages of temperature, the so-called normal temperatures, for numerous parallels of latitude published between 1852 and 1913 by Dove [

The poor coverage of the Southern Hemisphere by observations during that time indicated by the normal temperatures of Dove [

On the contrary, the average for the Northern Hemisphere of Dove [

To estimate the uncertainty of our results, the zonal averages of temperatures for all parallels of latitude were randomly perturbed (without presupposed seed) by adding temperature values, Δ R T ,that are normally distributed with a standard deviation of σ = ± 2 K . For each of these historical datasets, ensembles of the 50 realizations of perturbed distributions were created. The numerical integrations of these perturbed distributions provided uncertainties in the global averages ranging from ±0.3˚C to ±0.6˚C where the magnitude of uncertainty increases with the decreasing number of normal temperatures available. The global and hemispheric means obtained from the ensembles of perturbed distributions well agreed with those derived from the original unperturbed datasets.

To assess the difference between spherical and spheroidal averaging special attention was paid to the distributions of climatological mean temperatures for numerous parallels of latitude published by von Bezold [

Compared with our results, the hemispheric averages for the Northern Hemisphere of Dove [

〈 T 〉 N H = 15.5 ˚ C ,Ferrel [

〈 T 〉 N H = 14.3 ˚ C ) for 1881-1910. The Berkeley record provided 〈 T 〉 ≅ 13.6 ˚ C and 〈 T 〉 ≅ 13.5 ˚ C for these periods, respectively. The NASA GISS records yielded 〈 T 〉 ≅ 13.6 ˚ C ( 〈 T 〉 S H = 13.0 ˚ C and

〈 T 〉 N H = 14.2 ˚ C ) for 1881-1910. Obviously, these results are notably lower than those calculated from the meridional distributions of historical zonal averages of temperature. Since the HadCrut4 record yielded

〈 T 〉 = 14.4 ˚ C ( 〈 T 〉 S H = 13.7 ˚ C and 〈 T 〉 N H = 15.2 ˚ C ), the Berkeley record 〈 T 〉 = 14.5 ˚ C ,and the NASA GISS

records 〈 T 〉 = 14.5 ˚ C ( 〈 T 〉 S H = 13.7 ˚ C and 〈 T 〉 N H = 15.2 ˚ C ) for 1991-2018, the results derived from the

historical data suggest no change in the globally averaged near-surface temperature over the past 100 years.

Our results underline that reviewing the epoch-making literature from the 19^{th} century and the first two decades of the 20^{th} century is indispensable in the assessment of climate change since the end of the Little Ice Age in the first half of the 19^{th} century.

We thank the anonymous reviewers for fruitful comments. We thank Google Books and the Royal College of Physicians of Edinburgh for making the textbooks of von Hann and Börnstein and numerous reports available to us. We also thank ETH-Bibliothek Zürich for making the Atlas of Meteorology: a series of over four hundred maps prepared by John G. Bartholomew and Andrew J. Herbertson and edited by Alexander Buchan available to us as well.

The authors declare no conflicts of interest regarding the publication of this paper.