Note on the number of numbers less than a given Three parabolas connected with a plane triangle, 79 118 TWEEDIE, C. On the solution of the cubic and quartic [Title], 113 WALLACE, W. 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. 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. 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 To draw a straight line R1 R2 To put a point of the compasses on a determinate To put a point of the compasses on an indeter- To describe a circle 02 C2 No account is taken of the length of the lines that are described; if any portion of a straight line be drawn the operation is R2, if a small are only or the whole circumference be described, the operation is C3. It ought also to be added that to place the edge of the ruler in coincidence with two points is 2R1; to put one point of the compasses on a determinate point and the other point of the compasses on another determinate point is 2C1. Every construction therefore is finally represented by R2+R+m, C1 + m2С2 + m2С where l1, m1, etc., are coefficients denoting the number of times any particular operation is performed. The number (l + l2 + m2 + m2 + m) is called the coefficient of simplicity, or more shortly, the simplicity of the construction; it denotes the total number of operations. The number +m+m2 is called the coefficient of exactitude, or more shortly, the exactitude of the construction*; it denotes the number of preparatory operations, on which and not on the tracing operations, the exactitude of the construction depends. The number of straight lines drawn is l2; the number of circles mg. An objection at once presents itself to the reader, as it did to Mr Lemoine. Is it legitimate to suppose the operations R1, R, C1, C2, C3 identical in value, in order to make up the coefficient of simplicity or exactitude? They are evidently not identical in execution, and hence Geometrography does not furnish us with an absolute measure of simplicity or exactitude in the sense in which measure is usually employed, the comparison of one magnitude with a unit of the same kind. The various operations however are assimilated because they are incapable of decomposition into others more simple, and because, speculatively, any one is neither more simple nor less simple than another. In one case it may be said that Geometrography does furnish an absolute measure, the case namely when all the coefficients in one construction are smaller than the respective coefficients in the other. This case occurs pretty frequently. * Mr Lemoine remarks that the simplicity and the exactitude of an operation vary inversely as the numbers he sums; but since no confusion is possible, he prefers names recalling the object aimed at to the more logical terms coefficient of complication and coefficient of inexactitude. Such is Mr Lemoine's scheme of comparison, which he applies to more than sixty of the principal problems of elementary geometry, with some very unexpected results. To justify his procedure in denoting by 2R, and 2C, the operations of placing the edge of the ruler and the two points of the compasses in coincidence with two given points, Mr Lemoine says in a note on the problem To take with the compasses a given length AB: “It is clear that the operation of putting the first point of the compasses on A is not the same as that of keeping the first point on A and placing the second on B; and yet we denote them both by C1. We believe that there is no inconvenience in that, because we are only making an ideal theory of operations. Thus we suppose that all the lines of the figure intersect within the limits of the drawing, that it is indifferent whether these lines intersect at a very acute angle, and so on; so that it appears to us quite sufficient to denote by the symbol C, the general operation which consists in putting one of the points of the compasses on one point. The reader however who, after reflection, does not share our opinion has only to denote by C,' the operation which consists in putting on a given point the movable point of the compasses while the other is kept fixed. 1 “In like manner, since we call R, the operation which consists in putting the edge of the ruler in contact with a point, it is evident from the manner in which it is performed that the operation which consists in putting the edge of the ruler in coincidence with two given points is not exactly twice the operation R1. One might also denote by R1+R' the operation which consists in placing the edge. of the ruler in contact with two points; but if one practises Geometrography a little I believe he will come to recognise that this distinction is a useless complication. "We might also have assimilated the operations C, and C, and have kept for the two only one symbol C1; but we have not done so, because if theoretically R, and R' come to the same thing, C, and C, are theoretically different. C2 however occurs much more rarely than the other symbols, and in general with a very small coefficient." I am not sure that I understand in what respect the practical |