An alternative form is got by performing the multiplications and rearranging. It is ΣA2B2C2-2ΣA3BC-ΣA3BCD + 4Σ A2B2C-6ΣA2BCD The symmetry with respect to A, B, C, D is evident à priori. LXIII. A Mode of Calculating a Limit to the Direct Effect of Great Eccentricity of the Earth's Orbit on Terrestrial Temperatures, showing the Inadequacy of the Astronomical Theory of Ice Ages and Genial Ages. By EDWARD P. CULVERWELL, M.A., Fellow of Trinity College, Dublin*. Tin fus womental was writings on the Glacial Periods, and HE fundamental assumption made by Dr. James Croll subsequently adopted by Sir Robert S. Ball in 'The Cause of an Ice Age,' is that we may attain to some approximate idea of the lowering of terrestrial temperatures, due to greater winter distance from the sun, by the following considerations: Were it not for solar heat the earth would sink to what Ball calls its natural zero, which must be nearly the absolute zero of temperature. Hence the effect of the sun-heat is to maintain it at its present excess above that temperature, and any decrease in sun-heat will be accompanied by a more or less proportionate decrease in the excess of the earth's temperature above the natural zero. To be on the safe side, however, Croll takes this natural zero as a temperature of -239° F. (Pouillet's temperature of space), and Ball as -300° F. Croll, in chap. xix. of Climate and Time,' which he devotes to this subject, supposes the midwinter temperature in an epoch of great eccentricity to be proportional to the sun-heat received on midwinter day—a supposition open to the obvious criticism that the adjustment of temperature to sun-heat can hardly be instantaneous. This is probably the reason why Ball modified the argument and takes the average winter temperature as proportional to the average daily winter heat from equinox to equinox. I shall show, by actual comparison with terrestrial temperatures, that this assumption of the proportion between the *Communicated by the Author, temperature at a place and the sun-heat received is utterly wide of the mark; but first I will endeavour to trace the origin of Croll's mistake. In 1830 Herschel wrote a paper "On the Astronomical causes which may influence Geological Phenomena," and in it he dealt with the possible changes of terrestrial temperature due to changes in the earth's minor axis, making use of the argument that a small percentage alteration above the absolute zero (which he says some place at -1000° F., some at -5000° F., and some lower still) due to a small percentage change in the annual quantity of sun-heat received would produce all the great change in temperature required by geologists. In such a case the method is perfectly valid, except that terrestrial radiation should not be taken as simply proportional to absolute temperature. In the centuries during which the earth receives less annual heat from the sun we may fairly suppose that a practically permanent state is reached in which the heat annually received by the globe as a whole is equal to that radiated by the globe as a whole. In that case we cannot complain of Herschel in 1830 treating the radiation as proportional to the temperature, and then treating the percentage decrease in temperature of the globe as a whole as equal to the percentage decrease in sun-heat received. Had he taken the law of cooling as at present stated-i. e. radiation varies as the fourth power of absolute temperature-the percentage decrease in temperature would (for small changes) only be one fourth of the percentage decrease in heat received. Subsequently Herschel, in a rather confused paragraph in his 'Outlines, 369a, referred to by Croll (p. 37) as his authority, appears to apply the same method to calculating temperature differences between the northern and southern hemispheres in the period of great eccentricity. But he rejects the calculation immediately after on the ground that loss of heat in winter, through greater distance from the sun, is compensated for by greater duration of winter. Hence the paragraph does not really seem to justify Croll's application. For Croll applies the method to the temperature of limited portions of the earth's surface, in which, owing to the fact that heat is continually being transferred from one region to another, there is never a time at which the gain by sun-heat is equal to the loss by radiation. The statement just made shows that for Croll's argument to have any practical validity as a method of calculating the winter temperatures in the epoch of great eccentricity, it is necessary not only that the temperature adjustments shall be made with great rapidity but also that the direct effect of sun-heat on terrestrial temperatures in the latitudes with which glaciation is concerned shall be great compared with that of the heat transferred from latitude to latitude by air and ocean currents. That this is not the case in regard to the changes of temperature from summer to winter, we have a ready means of showing; indeed it appears that the direct effect of winter sun-heat in northern latitudes is but a trifling factor in keeping the temperature above the "natural zero" when compared with the heat transferred from place to place. For consider the equator: it receives nearly the same daily sunheat during winter and summer, and its temperature stands continuously about 380° F.above Ball's "natural zero." Hence, if we calculate and then plot on a diagram the average daily summer and winter sun-heats for each latitude, on such a scale that the equatorial sun-heat is represented by 380, then the ordinates giving the summer and winter sun-heats for the different latitudes should also, if Herschel's principle were applicable, give the summer and winter excesses of their temperatures above the natural zero. In fig. 1 this has been done for latitudes 40° to 90°, the only ones we are concerned with in dealing with the Glacial Period. I have also shown by the dotted lines the mean of the Challenger's' January and July temperatures for the meridians 90° E. and 90° W. passing through the middle of the great Asiatic and American continents up to latitude 70°, beyond which reliable information cannot be obtained. The scale of sun-heat is shown at the left side, and the scale of Fahrenheit temperatures at the right of the centre line. By examining the figure we see that even in the Continental areas, where the most extreme variations are observed, the midwinter temperatures bear no relation whatever to the very small amount of winter sun-heat; so that we infer that the transference of heat from one latitude to another is the chief factor in winter temperatures. If we were to follow Croll's method, and take the midsummer and midwinter sun-heats instead of the daily average, the discrepancy shown in the figure between actual temperatures and sun-heats would be enormously increased. To complete the criticism, let us examine the most extreme of all climates, that of Yakutsh in Siberia. Here the midsummer temperature rises to about 370° over the " natural zero and the midwinter sinks to 260°. But the daily average sun-heat in summer is to the daily average in winter as 1300 is to 260, or as 370 is to 74. Hence if the midwinter temperature be right, the midsummer ought to be 1000° F., or if the midsummer be right, the midwinter ought to be 226° F. If, following Croll's method, we calculate the midsummer and midwinter sun-heats, we find the ratio to be 7600 to 260, or 370 to 12.5. It is unnecessary to dwell on the discrepancy between these figures and the temperatures. Fig. 1.-Showing the want of proportionality between winter A A, curve of average daily summer sun-heat. a a, curve of mean Continental July temperatures. bb, curve of mean Continental January temperatures. When, therefore, there is such a complete absence of proportion between the summer and winter temperatures and the summer and winter sun-heats, what ground can be assigned for the assumption of a rough proportion between the changes in the summer or winter temperatures and the changes in the corresponding sun-heats due to eccentricity? Even if the temperatures themselves did show a certain rough proportion to the sun-heats, we could not expect as near an approach to the proportion in their changes, because the disturbing element might be large compared to the changes in the temperatures, though small compared to the temperatures themselves. As it is, the argument breaks down completely; and the assumption on which the supporters of the Astronomical theory of the Ice Ages and Genial Ages rely is found to be nothing but a vague speculation, not only unsupported by the physical evidence bearing on the subject, but in direct opposition to it. The weakness of the foundation, however, has not deterred the astronomical geologists from erecting on it an edifice of vast proportions. They affirm that there have been a number of successive glaciations, alternating with genial ages at intervals of about 10,500 years, that glaciation must have been simultaneous all round the pole, that a glacial age in the northern hemisphere must have been accompanied by a genial one in the southern, and vice versa. And as M. Leverrier's calculations show that the last period of considerable eccentricity was about 100,000 years ago, the last Ice Age must have occurred about that date. Dr. Croll indeed fixes all the possible dates for three million years of past and one million of future time; but Sir Robert Ball declines to fix the dates, not from any want of implicit confidence in the theory, but merely from a probable want of accuracy in the determination of the astronomical constants involved. I imagine, however, that Sir Robert Ball would accept as fairly reliable Leverrier's calculations for this comparatively short period, though he does not accept those of Croll for the longer period. Hence we may safely assume with Croll, that the last Ice Age must have terminated 80,000 years ago, if his theory be correct. When Croll's theory was originally published, the general opinion of geologists was that the date of the Ice Age as fixed by the theory was far too recent. But the uniformitarian school of those days has now been replaced by the modern school, which, relying on calculations of the rate of denudation, tends to fix the glaciation of North America as hardly more than ten or twenty thousand years ago. Hence geologists are now disposed to welcome any criticism of |