The object of the present paper is merely to exhibit the methods employed by the ancient Greek geometers in their solution of this celebrated problem. A critical discussion of these methods, of the origin of the problem, and perhaps also an account of more recent researches and a notice of the literature connected therewith may form the subject of a subsequent paper.

ABC is a triangle, right-angled at A, and X, Y, Z are the centres of the squares described on the sides opposite the angles A, B, C; XD, XM, XN are respectively perpendicular to BC, CA, AB; MY, DY are joined, and DY meets AC at E; NZ, DZ are joined, and DZ meets AB at F.

Now that more science has become the popular educational cry, there is a danger of raising too great expectations of what physical science can do, and so of paving the way for a reaction against it when it is found not to yield the results unduly expected of it. This arises mainly from basing the claims of physical science to a school place upon an exaggerated estimate of the value of the knowledge imparted, and from not admitting it as an educative agent capable of filling a unique place in the educational course. On account of the popular belief in the ultimate practical, or bread-and-butter value of science teaching, it has been introduced in many cases as an attraction in a school prospectus without adequate means being provided for efficiently carrying it on. In such cases it often takes the form of the popular lecture illustrated by experiments which requires no great mental application on the pupil's part, gives him amusement, and relieves him for the time from some of his dry daily routine, but which fills him at the same time with a false and mischievous notion of what science is. It is of great importance that the true place and aim of physical science in schools should be clearly recognised, not merely by scientific men and educationists, but also by the intelligent general public, for it is only when such recognition is general that the means of equipping and maintaining science work will be forthcoming.

Let l be the length of the rectangle and b its breadth, while s is the side of the square equal to it in area; s is found, of course, by taking l: s = s : b. A number of different cases arise.

This paper was based mainly on the results of an investigation which will appear in full in the Transactions of the Royal Society of Edinburgh. Incidentally, however, it led to a discussion of the question:—Find the law of density of a planet's atmosphere, supposing Boyle's law to be true for all pressures, and the temperature to be uniform throughout.

1. The word contour is largely used in ordinary language, but its meaning, when so used, is in general very different from its meaning as a scientific term. We speak of the contour of a hill, a cloud, a country, and so on; meaning usually a profile or an outline,—sometimes a particular outline only. Yet, even in this popular use of the word, we have an indication of its more exact significance. Thus, we see that the visible horizon, if we consider it to be a contour line, is the curve in which the earth's surface is met by its tangent-cone the vertex of which is at the observer's eye. The tangent-surface has a constant characteristic; and it is this possession of a distinctive property by all surfaces which give rise to contour lines, which furnishes the reason for the peculiar applicability of the method of contours to physical problems.

For the measurement of a plane area, bounded by an irregular curve, various methods are adopted. Besides the well-known methods of approximation in use among land measurers, the following may be mentioned—

The area is divided by parallel ordinates. These are measured, and the results are then treated in several ways to produce more or less accurate results. See Williamson's Integral Calculus, third edition, p. 211.

The problem is to show that the expression

where ω is an imaginary fourth root of unity, is symmetrical with respect to a, b, and c.

§1. Figure 30. Let AB be any straight line in the plane figure, A′B the position of the same line after a displacement, the point A moving to A′ and B to B′. Since the position of a plane figure in its plane is determined, when the position of a straight line rigidly attached to it is determined, the motion of the plane figure is determined if we determine the motion of the straight line AB.

If y=z + xy(y) where x and z are independent, Lagrange's series gives the expansion of any function of y in terms of x. The coefficients of the powers of x may be found thus. Let ø(y) be the function to be expanded. Then by Taylor's theorem

The object of the following paper is to illustrate how readily some problems, which frequently occur in physical work, may be solved by the application of certain methods which are not very generally employed in mathematical investigations. The method on which the following proof is based is that of contour lines. These are a device enabling us to represent the third dimension on a plane, and are such that all the points on one contour line are at one and the same height above the level of the base plane. Probably the best known application of the principle of contours is to be found in the Survey Maps, where successive (closed) curves pass through all points at heights of 100, 200, 300, &c., feet above the sea-level; but they are of equal value in many physical diagrams, where they appear as equipotential lines, isothermals, lines of equal illumination, lines of equal force in a magnetic field, as well as in many other forms. In this paper they are employed as representing the third dimension merely. When contour lines are placed so as to indicate an equal rise in the intervals between each successive pair [or, in fact, according to any definite system], we can, with their aid, tell two things about a surface; the one—what is the steepness, or gradient, in any direction from a given point; the other—which is the direction of steepest slope at any point. The absolute steepness is measured by the amount of vertical motion per unit of horizontal motion of a point; i.e., by the tangent of the angle between the horizontal plane and the line in which the point moves. This is a fact with which everyone is acquainted ; every person knows what is meant by saying that the gradient along a line of railway is 1 in 50 or 1 in 500, as the case may be. To find the steepness at any point we merely need to know, in terms of the length of the line taken to represent unit steepness, the length of the line passing through the point in the given direction, and terminated by the two contours on either side of that point. The length of the line will then be inversely as the steepness.

The paper is entitled Solutio facilis problematum quorumdam geometricorum difficillimorum, and is printed in Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae, Tom. xi., pp. 103–123. The volume is for the year 1765; the title-page is dated 1767.

Of the following properties the first two are obvious; the third was communicated to the Royal Society of London by the Rev. Dr James Booth* in 1854; the others, obtained many years ago, have not as far as I know been remarked. They seem to be more curious than useful.

If A and B be two fixed points on a great circular arc and P a variable point on the arc, there are two and only two possible positions of the point P corresponding to a given valae of the ratio sinAP/sinBP, provided arcs measured in one direction from A or B be considered positive, and in the opposite direction negative; and these two points are antipodal.

1. Figure 54. If ABC be a triangle, P any point, then the system of forces PA, PB, PC is equivalent to the system PH, PK, PL, where H, K, L are the middle points of BC, CA, AB.

[The foflowing method of dividing a straight line in medial section was communicated to Mr Munn of the Edinburgh High School, by the Right Hon. Hugh C. E. Childers in January last.]

Let AB be the line. (Fig 56).

Draw AC = ½AB, and at right angles to it. Join CB. Bisect the angle ACB with CD, cutting AB at D. Draw DE at right angles to CB. Then the triangles CAD, CED are equal, and AD = DE. Draw a circle with DA and DE radii, cutting AB at F.

1. In the first part of this paper we have considered merely the contours of curves, that is, contour points, and the method of obtaining the various physical diagrams. In this part we shall consider chiefly the contours of surfaces; that is, contour lines.

If any curve be cut by planes parallel to that of (x, y) and if the various points of intersection be projected on any one of these planes, say z = 0, the contour points so obtained will evidently lie on a definite line, and the line will be more accurately indicated in proportion as the number of intersecting planes is greater and their mutual distance is less. It will be given without any break in continuity by projecting every point of the curve upon the plane z = 0. But such a line may be regarded as the intersection, by the plane z = 0, (see fig. 48) of a cylindrical surface whose generating lines are parallel to the z-axis and are drawn from the given curve to meet that plane. We have here then the intersection of a given surface by a surface over which z is constant. But this satisfies our definition of a contour line. This case of a cylindrical surface supplies the simplest system of contour lines by giving z different values. The contours are all superposed in the diagram, but are not in general conterminous. The only case in which they would be conterminous is that in which the same values of the x and y co-ordinates of a point on the curve correspond to different values of the z-co-ordinate.

Certain cases of motion in a gas contained in a spherical or in a cylindrical envelope or surrounding a sphere or a cylinder have been considered by Professor Stokes and Lord Rayleigh, of which a clear idea may be obtained from chapters xvii. and xviii. of Rayleigh's Sound. Their chief object is to determine the motion in the gas when the motion in the bounding surface is given, or to determine the modification in the nature of a wave being propagated through the gas by the presence of an obstacle.