Arrhenius had some good data to observe. It had already been noted by Frank Washington Very and Samuel Pierpoint Langley that the infrared radiation from the moon which was observed at the surface of the earth changed its characteristics according to the angle of the moon above the horizon. These observations agreed with the properties of infrared absorption by carbon dioxide observed by Tyndall. (This also had a very important implication, which was missed by everyone, including Arrhenius, for more than 50 years after Arrhenius published his greenhouse theory. More on this later.)
Arrhenius also had a new analytical tool--the Stefan-Boltzmann law which had been deduced by Jožef Stefan in 1879, with some refinements added in 1884 by his student Ludwig Boltzmann. Arrhenius needed these mathematical tools to do his calculations. In theory, it was simple to calculate. Arrhenius was smart enough to realize it would not be so simple.
Aside from the stress coming from his divorce, which his wife Sofia Rudbeck was making as difficult as possible (this being the late 19th century, when divorce was strongly frowned upon and scandalous in itself) and teaching students and grading papers, which Arrhenius could do on autopilot, there was not much else to do. Arrhenius threw himself into the problem.
First he formulated his 'greenhouse law' (which still stands the test of time.)
If the quantity of carbonic acid increases in geometric progression, the augmentation of the temperature will increase nearly in arithmetic progression.
This simplified expression is still used today:
ΔF = α ln(C/C0)
In short, each doubling of carbon dioxide in the atmosphere increases global warming by the same amount. A rise from 300 ppm to 600 ppm will increase global temperature the same amount as a rise from 600 ppm to 1,200 ppm will.
Arrhenius also took into account the effect that decreased snow cover would have on the albedo (reflectivity) of the Earth--showing how the feedback of decreased snowcover (or increase if carbon dioxide declined and the temperature fell) would augment the effects of changed in the carbon dioxide concentration. Arrhenius also took into account the effect that a warmer atmosphere would have on water vapor in the atmosphere---a warmer atmosphere could hold more water vapor--and water vapor is a potent greenhouse gas in itself. Figuring out how these feedbacks would augment each other was complex. He spent all of 1895 in tedious calculation, which gave him something to do while going through his divorce, and published On the Influence of Carbonic Acid in the Air upon the Temperature of the Ground in the London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science in the April 1896 issue here.
Arrhenius' theory of anthropogenic global warming made a big splash. It was vigorously debated at the time (and still is, of course.) Knut Ångström made a strong criticism using the results of his experiments on the absorption of infrared radiation by carbon dioxide--which indicated that carbon dioxide was at a high enough level already that it absorbed all the infrared radiation that it could. Ångström and Arrhenius battled it out, but most physicists considered the Ångström criticism to be valid.
It later turned out that while CO2 absorbs all the IR radiation it can in two bands, CO2 absorbs in other bands as its concentration increases. Ångström had made his measurements with accurate equipment, but in a laboratory at room temperature. He didn't try matching the conditions one would see in the polar or upper level environments. As it turns out, in the polar regions and at dry mid and upper levels of the atmosphere there is not enough CO2 to absorb infrared radiation even in the two primary bands.
And there was another clue, staring people in the face---although no one realized it for over 50 years. When Frank Washington Very and Samuel Pierpoint Langley took their measurements of infrared radiation from the moon, the absorption increased when the moon was close to the horizon, and decreased when it was high in the sky. That showed right there that the absorption of infrared radiation was not saturated---otherwise it would have been the same---saturated in every direction equally! But no one realized this!
Arrhenius fought Ångström hard on the infrared absorption issue, but there was another problem. The ocean is alkaline, and was believed to absorb CO2 (carbonic acid) efficiently. In other words, we could emit CO2 into the atmosphere, but it would be mopped up in the ocean. The chemistry of the ocean was being determined and it was apparent that there was about 50 times as much CO2 in the oceans as in the atmosphere. Surely the ocean could absorb what CO2 we emitted easily!
Arrhenius was doubtful about that---although he admitted the oceans could have a buffering effect. Arrenhius' main arguement was that the oceans would not absorb CO2 immediately---and there was also the fact that despite the oceans are alkaline, there is still CO2 in the air? Why wasn't all atmospheric CO2 absorbed completely?
There must be some mechanism that keeps CO2 in balance and prevents its complete absorption by the oceans. And that mechanism could prevent all additional CO2 mankind emitted from being absorbed by the seas.
But this was a weak argument without chemical knowledge and proof of such a buffering effect, and Arrhenius knew it. Arrhenius had strong support from Alfred Wallace in that there must be a buffering effect from the sea, despite the seas' alkaline nature. But it was not until the late 1950s and early 1960s that the buffering effect was shown to exist---and that came about because the rising CO2 levels documented by Charles Keeling in the first few years of his measurements defied explanation otherwise. (there was some work in the late 1950s that suggested that CO2 absorption by the oceans was buffered, but it was not until Keeling's graph that a strong research effort was done).
Arrhenius also publicized another radical idea--panspermia---the spread of life from planet to planet by spores. As is usually the case, panspermia was an idea that had been kicked around by others before him---but as a speculative idea--not as a serious one. He first wrote about panspermia in 1903, and made it a major part of his book for educated popular audiences, Worlds in the Making--the Evolution of the Universe The original English translation is from 1908 (which I have linked). The translation is a bit rough and stilted, since the book was originally published in Swedish as Världarnas utveckling in 1906 and translated into German as Das Werden der Welten (1907). The English translation is from the German--and it would be interesting to see if the book is available translated directly into English in more modern language. We could get Grothar on it ;)
Worlds in the Making--the Evolution of the Universe also talks about the anthropogenic greenhouse effect. Some extracts follow:
"To a certain extent the temperature of the earth's surface, as we shall presently see, is conditioned by the properties of the atmosphere surrounding it, and particularly by the permeability of the latter for the rays of heat." (p46)
"That the atmospheric envelopes limit the heat losses from the planets had been suggested about 1800 by the great French physicist Fourier. His ideas were further developed afterwards by Pouillet and Tyndall. Their theory has been styled the hot-house theory, because they thought that the atmosphere acted after the manner of the glass panes of hot-houses." (p51)
"If the quantity of carbonic acid in the air should sink to one-half its present percentage, the temperature would fall by about 4°; a diminution to one-quarter would reduce the temperature by 8°. On the other hand, any doubling of the percentage of carbon dioxide in the air would raise the temperature of the earth's surface by 4°; and if the carbon dioxide were increased fourfold, the temperature would rise by 8°." (p53) [these are degrees Celsius, not Fahrenheit]
"Although the sea, by absorbing carbonic acid, acts as a regulator of huge capacity, which takes up about five-sixths of the produced carbonic acid, we yet recognize that the slight percentage of carbonic acid in the atmosphere may by the advances of industry be changed to a noticeable degree in the course of a few centuries." (p54)
*note--it turns out that the oceans absorb only half of CO2 we emit, not 5/6ths
"Since, now, warm ages have alternated with glacial periods, even after man appeared on the earth, we have to ask ourselves: Is it probable that we shall in the coming geological ages be visited by a new ice period that will drive us from our temperate countries into the hotter climates of Africa? There does not appear to be much ground for such an apprehension. The enormous combustion of coal by our industrial establishments suffices to increase the percentage of carbon dioxide in the air to a perceptible degree." (p61)
The last quotation from Worlds in the Making--the Evolution of the Universe, seen below, is interesting. Arrhenius thought that the anthropogenic greenhouse effect would be a benefit to mankind. Vast areas of Canada and Russia would be open to cultivation. Arrhenius correctly predicted that global warming would be more pronounced at the poles than in the tropics, so tropical life would not be affected greatly, he felt. The consequences of ice sheet melt and sea level rise do not seem to have entered his mind. Perhaps it was two factors that limited his foresight.
First, he was a Swede. Sweden is a cold place, where summers are pleasant, brief, and looked forward too. Arrhenius thought that places like Sweden would benefit from global warming.
Second, Arrhenius seriously underestimated how much the consumption of coal would increase---and he ignored oil and natural gas, which were not being used much at the time. By the 1890s, we knew about how much CO2 was in the air---measurements had narrowed it down to between 250 ppm and 350 ppm---300 ppm was what Arrhenius estimated. And that figure was close to correct at that time. However, by missing the growth in the consumption of fossil fuels, Arrhenius underestimated badly how rapidly CO2 would rise. He thought it would take 3,000 years for CO2 to double to 600 ppm. Instead we will do it before the 21st century is over--less than 200 years. A period of 3,000 years would be easier to adapt to, with so much more time. And as it turned out, Arrhenius was right about the oceans not absorbing all the CO2 humankind emits. But he underestimated there too. The oceans absorb only half the CO2 we emit, not 5/6ths. But we need to give Arrhenius credit for realizing that not all CO2 that we emit would be absorbed by the seas.
The last quote is below:
"We often hear lamentations that the coal stored up in the earth is wasted by the present generation without any thought of the future, and we are terrified by the awful destruction of life and property which has followed the volcanic eruptions of our days. We may find a kind of consolation in the consideration that here, as in every other case, there is good mixed with the evil. By the influence of the increasing percentage of carbonic acid in the atmosphere, we may hope to enjoy ages with more equable and better climates, especially as regards the colder regions of the earth, ages when the earth will bring forth much more abundant crops than at present, for the benefit of rapidly propagating mankind." (p63)
It has to be said that Arrhenius was not a person with great integrity. In 1900 he became part of the Nobel Prize Committee on Physics, and the de facto head of the Nobel Prize Committee on Chemistry. Arrhenius basically awarded the 1903 Nobel Prize for Chemistry to himself! And he used (abused) his position to award Nobel Prizes to his friends (Jacobus van't Hoff, Wilhelm Ostwald, Theodore Richards) and to deny Nobel Prizes to his enemies (Paul Ehrlich, Walther Nernst). Both of the later did receive Nobel Prizes, but Paul Ehrlich never got a Nobel Prize in Chemistry, being granted a Nobel Prize for Medicine instead. Walther Nernst received a Nobel Prize in 1920 after Arrhenius blocked it for 20 years---his achievements in chemistry were so notable that the rest of the Nobel Committee revolted in 1920 to award him the prize. Arrhenius definitely had a vindictive side to him.