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Life and Letters of Charles Darwin, volume 2 includes the following letter from Darwin to Hooker, about a examining a plant that absorbs ammonia:-

Reflect how little ammonia a plant can get growing on poor soil—yet it is nourished. The really surprising part seems to me that the effect should be visible, and not under very high power; for after trying a high power, I thought it would be safer not to consider any effect which was not plainly visible under a two-thirds object glass and middle eye-piece.

To which thing does "two-thirds" refer here, i.e the size or magnification or something else?

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The use here of "object glass" is equivalent to our current usage of "objective lens" on a microscope. In this case Darwin used a 2/3 lens and the "middle eye-piece" from a particular microscope. I think (but can't find confirmation) that the 2/3 refers to the width of the field of view in inches.

The letter you quote is to Edward Cresy Jr. on the 2nd November 1860. I can't find much on Darwin's microscopes, other than in the Whipple Museum, where they have 2 microscopes of Darwin's. One of these purchased in 1847 is a microscope with exchangeable lenses. This microscope was a Smith and Beck compound microscope. You can find examples of these for sale in antiques collections and a similar one is probably this one, which comes with a number of objectives and a 3 magnifications of eye-pieces (~1/2 way down page). The magnification of a 2/3 objective looks to me to be about 10-20x (confirmed in this review as ~15x (about 2/3 way down page)).

As for the middle eye-piece, based on the same Microscopy UK page linked above, it is probably a #2 eye-piece of about 8.5x, so a total magnification of about 125x (127.5x, if the numbers are accurate).

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