History of diatom microscopy
Diatoms and most other unicellular algae were not discovered until the invention of microscopes. The earliest description that we can be confident does refer to a diatom dates from 1703. It is contained in the Philosophical Transactions of the Royal Society of London and was in the second of 'Two letters from a Gentleman in the Country, relating to Mr. Leeuwenhoek's Letter in Transaction, no. 283.'' [Anonymous, Phil. Trans. R. Soc. Lond. 23 (288): 1494-1498 and 1498-1501, November/December 1703]. The letter was communicated to the Society by 'Mr C.', who was presumably a Fellow.
In the first letter, the anonymous microscopist says he had received Transaction 283 and 'Finding something in it from Mr. Leeuwenhoek, I greedily run it over, and must own that I was a little mortified to see in it an account of a Creature which I thought I had a sort of Propriety in, and of which I had made a Draught, with a design to present you and Mr C. with a rarity, which I believed no body had met with but myself.' However, Leeuwenhoek had seen the organism - a Hydra - first and so the Country Gentleman was pre-empted. But he thought that he could provide a better illustration, and indeed he did. In the same letter, he talks also about the 'Lens Palustris' or 'Duck-meat' - his name for Lemna - and mentions that it comes from the bottom (as we know now, from turions).
In the second letter, the Country Gentleman describes further observations of 'Lens Palustris' roots and notes that 'I often saw adhering to them (and sometime separate in the Water) many pretty branches, compos'd of rectangular oblongs and exact squares, which were joyn'd together ... There are often twenty or more of these Figures in one branch, which generally adheres at one end to the Stalks of the Plant, and I think it remarkable that these regular parallelograms are all of the same size, the longest side not exceeds 1/3 of an hairs breadth, and that the length is just double the breadth, the squares being visibly made up of two parallelograms joyn'd longwise. They seem very thin and the texture of every one is nearly the same... I took these branches at first for Salts, but finding them always of the same size, and that there was no sensible encrease of their bulk while they continued in the Water, that after they had lain a day or two dry on a Glass Plate they alter'd not their Figure, and upon the addition of new Water (warm or cold) they had still the same appearance and cohesion, and that their adherence (tho touching only in the angular points) was so firm and rigid, that all mov'd together, and kept the same position in respect of one another, however agitated by the Water; these considerations, I say, persuade me, that they may be rather Plants than Salts, but they being so very minute that no judgment can be made of 'em but by the Eye, I shall not determine any thing positively.'
The illustrations given in the 1703 paper make it almost certain that the Gentleman was studying Tabellaria. Notice his remarkable insight that the structures were most likely plants, despite the fact that no plants known at that time were remotely similar to Tabellaria. The shallow depth, firm connection between cells, and quadrilateral geometrical shapes are all characteristic of Tabellaria, and maintenance of shape despite desiccation and subsequent rehydration reflects the silica cell wall that is the diatoms' trade-mark.
I have not been able to determine what kind of microscope the Country Gentleman used. His contemporary and rival, Antonie van Leeuwenhoek, used single lens microscopes, which were extremely simple, but highly effective in Leeuwenhoek's hands. The glass lenses were spherical - either blown or produced by grinding beads - and could achieve a resolution of less than 2 µm. A full account is given by Brian J. Ford [The Leeuwenhoek Legacy. Biopress, Bristol and Farrand Press, London (1991), 185 pp.]. However, Leeuwenkoek's methods were not adopted by others and it is more likely that the Country Gentleman used an early compound microscope, following improvements in optics made by Robert Hooke in the seventeenth century. Leeuwenhoek almost certainly saw diatoms - they are present in some of his samples (Ford, op. cit.) - but his illustrations and descriptions are not detailed enough for us to be sure.
What these early developments show is how much can be done with primitive equipment. What is sad is that today's microscopists often don't do much better, despite the high quality and ease of use of today's instruments.