So having found a copy of the first edition of Modern English Usage on the shelves, I was pleased to notice the article
progression. Arithmetical p. & geometrical p. There are in constant demand to express a rapid rate of increase, which is not involved in either of them, & is not even suggested by a. p. Those who use the expressions should bear in mind (1) that you cannot determine the nature of the progression from two terms whose relative place in the series is unknown, (2) that ever rate of increase that could be named is slower than some rates of a. p. & g .p., & faster than some others & consequently (3) that the phrases 'better than a. p., than g. p.', 'almost in a. p., g. p.', are wholly meaningless.
In 1903 there were ten thousand 'paying guests', last year  fifty thousand. The rate of increase is better, it will be observed than arithmetical progression. Better, certainly, than a. p. with increment 1, of which the fourth annual term would have been 10,003; but as certainly worse than a. p. with increment a million, of which the fourth term would have been 3,010,000; neither better nor worse than, but a case of, a. p. with increment 13333 1/3. The writer meant a. p. with annual increment 10,000; but as soon as we see what he meant to say we see also that it was not wortth saying, since it tells us no more than that, as we knew before, fifty thousand is greater than forty thousand.
Even g. p. may be so slow that to raise 210,000 in three years to as little as the 10,003 mentioned above is merely a matter of fixing the increment ratio low enough. Neither a. p. nor g. p. necessarily implies rapid progress. The point of the contrast between them is that one involves growth or decline at a constant pace, & the other at an increasing pace. Hence the famous sentence in Malthus about population & subsistence, the first increasing in a g. & the second in an a. ratio, which perhaps started the phrases on their career as POPULARIZED TECHNICALITIES.