( « ) 
IV. — On Dr. Schumann's Formulte for Diatom-lines. 
By W. J. HiCKiE, M.A., S. John's College, Cambridge. 
Plate CIX. 
In compliance with a general and strongly expressed wish to that 
effect, we have here attempted to give the readers of the ' M. M. J.' 
some sort of an idea of Dr. Schumann's famous work, though we 
fear that, for so doing, some may liken us to the simple-minded 
man in Hierocles, who exhibited a brick as a specimen of the house 
EXPLANATION OF PLATE CIX. 
Fig. 1. — Amphipleura pellucida. " One specimen gave 45 transverse lines, and 
45 longitudinal lines, and 61 inclined lines in Yho'"- Another specimen exhi- 
bited 38 transverse lines and 55 inclined lines. If we add these, we get 
a = h = 41 J, G = 58. As 41^.^2 — 58|, there is no doubt that we have here 
corresponding series of the simplest sort." 
By inclined lines Dr. Schumann means imaginary lines drawn at a certain 
angle from a top row of transverse dots to a lower row of transverse dots. See 
his own explanation on pp. 8 and 13 of this number. 
Fia. 2. — Navicula lata (= Fiunularia pachyptera Ehrenh.). "Four specimens 
from the Siebenseethal exhibited 7 canals, and one from the Kohlbachthal gave 
12 canals in -r~"'. The closeness of the series of puncta traversing tiie whole 
frustule is defined by the formula a = 4 a." 
Fig. 3. — Navicula borealis. The closeness of its series of puncta is defined by 
the well-founded formula a = 4 a." 
Fig. 4. — Mastogloia antiqua Schum. " Whilst specimens found in Prussia gave 
27 transverse lines in 3^^"', tliose from the High Tatra exhibited 44, 45, and 48 ; 
to which estimates the formula 
" = - + 4-^ 
corresponds." 
Fig. b — Navicula crassinervis var. Compiire tlie figure of this in Smith's 
Synopsis. Dr. Schumann's figure gives the transverse lines alone. Pritchard 
(Infusoria) says, " Strise wtmting or indistinct." 
Fig. 6.— Navicula 7-homboides? (Ehrenb. Amer. III. 1, 15). "Three frustules 
from the Mengsdorfthal with 78 transverse lines in t;^'" appear to belong to 
Ehrenberg's form, though their length is only 8-9 T." His drawing gives no 
longitudinal lines. He seems therefore either to have overlooked them, or not to 
have known of their existence. 
Fig. 7. — Nav. rhynchocephala. "In one part of the High Tatra with 34, in 
another with 36 punctated transverse lines in xno'"* " 
Fig. 8. — Nkv. angastata. ''This diatom has in Prussia 40, and in my highe. t 
station on the Tatra 68 transverse lines in -roTr"'-" 
Fig. 9. — Campylodiscus nanus m. " On the margin appears a series of fine 
puncta, which, at an elevation of 5500 feet, numbered 53 in -1^"'. " 
Fig. 10. — Nitzschia minutissima. " On combining my estimates of ten speci- 
mens found in Prussia with my estimates of the sixteen found ou the Tatra, 
1 obtained the following formula for the number of marginal puncta : 
If this formula were well established, we sliould have an excellent means of 
(Ictcrmining the height of a place by the help of microscopical obiLervations, as 
this diatom is jirobably frequent on other mountain ranges also. ' 
