The following is my translation of an article called: 'Wieviel Tierarten gibt es auf der Erde?'. The author's name isn't given, but it first appeared in a German popular science magazine, Kosmos Handweiser für Naturfreunde 1910, Heft 4, Seite 152. I'm not aware of any previous translation.
As the information is in a compressed article without paragraphs, I summarised it for the sake of convenience in a table below.
Trevor Dykes.

How many species of animal are there on Earth?
At the British Association for the Advancement of Science, the well known zoologist, AE Shipley, has made a number of interesting reports on the progress of zoological research. About a decade ago, Günther presented a list of animal species, based on precise studies, as known in 1830 and 50 years later. Zoological science was aware of 1,200 different species of mammal in 1830; in 1881 the number of known species had grown to 2,300. Bird species in 1830 were put at 3,600; research knew of 11,000 in 1881. With reptiles, research strode from 543 to 3,400 (in the year 1881); the smaller the animal then the larger is the number of species meanwhile known. Between 1830 and 1881, the number of species of fish climbed from 3,500 to 11,000, the molluscs from 11,000 to 33,000, the bryozoen from 40 to 120, the crustaceans from 1,290 (1840) to 7,500, spiders and the like from 1,048 to 8,070, the millipedes from 450 to 1,300, the insects from 49,100 to 220,150, the echinoderms from 230 (1838) to 18,043, the worms from 372 (1838) to 6,070, the corals from 500 (1834) to 2,200, the poriferae from 50 (1835) to 400 and the protisten from 305 (1838-1844) to 3,500. In 1840 one knew of about 73,588 species of animal; in 1881 the number of researched animal species had already increased to 311,653. One would not go wrong should one assume that, every year since 1881, an average of 12,000 new species have been discovered, so that up to 600,000 species are known today. That is an enormous number, but it is only few in comparison to the as yet unknown extant species. In 1895, the zoologist Sharpe calculated the number of known insect species as being a quarter of a million, but he spoke of his conviction that these 250,000 insect species represented only a tenth of those existing on Earth which have as yet remained unknown to science. Precisely in the kingdoms of small animal species can be found a limitless and unbridgeable open field of research, while one can reasonably assume that with larger animals, especially the mammals, that most species are known to science.