A proof of the uniform convergence of the Fourier
series, with notes on the differentiation of the

112

26

On ridge lines and lines connected with them [Title],
On the highest wave of permanent type [Title],

Note on the number of numbers less than a given

number and prime to it,

The pedal triangle,

Three parabolas connected with a plane triangle,
Notes on an orthocentric triangle,

79

118

TWEEDIE, C.

On the solution of the cubic and quartic [Title],

WALLACE, W.

113

Note on a third mode of section of the straight line,

76

PROCEEDINGS

OF THE

EDINBURGH MATHEMATICAL SOCIETY.

TWELFTH SESSION, 1893-94.

First Meeting, November 10th, 1893.

JOHN ALISON, Esq., M.A., F.R.S.E., President, in the Chair.

For this Session the following Office-bearers were elected :

President--Professor C. G. KNOTT, D.Sc., F.R.S.E.
Vice-President-Mr JOHN M'COWAN, M.A., D.Sc.
Secretary Mr JOHN B. CLARK, M.A., F.R.S.E.
Treasurer-Rev. JOHN WILSON, M.A., F.R.S.E.

Editors of Proceedings

Professor KNott.

Mr A. J. PRESSLAND, M.A., F.R.S.E.

Committee.

Messrs J. W. BUTTERS; W. J. MACDONALD, M.A., F.R.S.E.; WM.

PEDDIE, D.Sc., F.R.S. E.; CHAS. TWEEDIE, M.A., B.Sc.; WM.
WALLACE, M.A.

The Geometrography of Euclid's Problems.

By J. S. MACKAY, M.A., LL.D.

The term Geometrography is new to mathematical science, and it may be defined, in the words of its inventor, as "the art of geometrical constructions."

Certain constructions are, it is well known, simpler than certain others, but in many cases the simplicity of a construction does not consist in the practical execution, but in the brevity of the statement, of what has to be done. Can then any criterion be laid down by which an estimate may be formed of the relative simplicity of several different constructions for attaining the same end?

This is the question which Mr Émile Lemoine put to himself some years ago, and which he very ingeniously answered in a memoir read at the Oran meeting (1888) of the French Association for the Advancement of the Sciences. Mr Lemoine has since returned to the subject, and his maturer views will be found in another memoir read at the Pau meeting (1892) of the same Association. The object of the present paper is to give an account of Mr Lemoine's method of estimation, to suggest a slight modification of it, and to apply it to the problems contained in the first six books of Euclid's Elements.

In the first place Mr Lemoine restricts himself, as Euclid does, to constructions executed with the ruler and the compasses, and these he divides into the following elementary operations :

To place the edge of the ruler in coincidence with
a point

R1

To draw a straight line

R2

To put a point of the compasses on a determinate
point

To put a point of the compasses on an indeter-
minate point of a line