90 THE ESSEX NATURALIST.
Norfolk," illustrating his remarks by a series of lantern slides and by actual
specimens which showed the methods of flaking adopted for the prepara-
tion of flints for the facing of church and other walls in mediaeval and
later times. Mr. Warren remarked on the resemblance of some of the
flint-debris to prehistoric implements.
After a discussion, the thanks of the meeting were accorded to Mr.
Warren for his lecture.
Mr. Avery gave a description of the twenty-three prints of Castle Hed-
ingham, Tilbury Fort, Upminster, etc., exhibited by him, and was voted
the thanks of the meeting for his exhibit.
The President called attention to some important recent researches
on the algae of the soil, particularly alluding to a paper which had been
read before the Linnean Society in March, 1918, which gave an account
of an investigation carried on by Miss Muriel Bristol, M.Sc. By culture-
methods the soil alga, Chlorococcum humicola, was made to grow abund-
antly, and biciliate zoogonidia were produced in large numbers. At the
reception by the President of the Linnean Society in October, 1921, Miss
Bristol exhibited cultures of soil-algae in agar + mineral salts + glucose,
which, even in the dark, had developed an intense, vivid green colour.
The President remarked that the results of such cultures opened up ques-
tions as to the nutrition required by soil-algae, and asked:—Is the alga
a saprophyte when colourless and buried in the soil?
At the conclusion of the President's remarks, Miss G. Lister proposed
the thanks of the meeting to him for his discourse, and these were heartily
accorded.
Mr. D. J. Scourfield then gave a lecture on "The Logarithmic Spiral
in Nature," which he illustrated by a series of lantern slides and by various
specimens from the Club's Museum, such as shells of Nautilus, Ammonites
and various Gastropods, opercula of Turbo, models of shells of foraminifera,
the cochlea of the human ear, etc.
Mr. Scourfield furnishes the following resume:—
THE LOGARITHMIC SPIRAL IN NATURE.
The subject of spirals in nature is a very large and extremely interest-
ing one; it is proposed on this occasion to confine attention to the par-
ticular form of spiral known as the logarithmic spiral. To the question—
"What is a logarithmic spiral?"—a provisional answer can be given
that it is that very beautiful expanding spiral with which all are familiar
in the discoid forms of univalve shells, such as a Planorbis or an Ammonite,
in the crozier of an opening fern frond, in the inflorescence of the forget-
me-not, and in many other natural objects. Mathematically it is the
spiral which is conceived of as being traced by a point moving with in-
creasing velocity along a line which is itself revolving uniformly round
one of its extremities, the velocity of the point at any moment being pro-
portional to its distance from the centre of revolution ("pole" of the
spiral). The appellation "logarithmic" is due to the fact that the angles
about the pole of such a spiral are proportional to the logarithms of the
successive radii. With the help of a table of logarithms it is therefore
easy to construct a logarithmic spiral having any desired ratio of increase.
Thus, assuming a threefold increase for each revolution and ten equal