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This booklet is a collection of the papers presented at a joint session of the Mathematical Association of America and the National Council of Teachers of Mathematics during the fiftieth annual meeting of the MAA. These papers were presented under the headings of "Geometry and School Mathematics,""High School Geometry," and "School Geometry and the Future." Eight papers are presented in their entirety. An abstract of a ninth paper, "Geometry: The Cambridge Conference View" by Edwin E. Moise is also included. Authors of the individual papers do not act as spokesmen for the MAA or NCTM but express personal viewpoints. (FL)

This document is the report of a curriculum writing session. Using the recommendations and suggestions of the planning conference (SE 012 733), the writers produced detailed outlines for the grade seven through nine curriculum. The purpose and rationale of each unit are stated, and the separate sections outlined. Sample exercises are included for most units. The aims of making mathematics more relevant to real world problems, and of avoiding excessive abstraction and rigor, are continually stressed. Also included are three general reports on geometry, real analysis, and mathematical models; several position papers on specific topics; and a comparison of the new syllabus with the "first round" curriculum. (MM)

Sixteen mathematicians from universities, schools and industries met for five days in 1966 to plan a "second round" curriculum which would place greater emphasis on the relevance of mathematics to problems of the real world. Their deliberations are reported in this document. The three main recommendations were that the curriculum should contain: (1) frequent consideration of mathematical models of significant problem situations; (2) an introduction to those mathematical concepts which are important to the general citizen; and (3) probability, logic, computing and flow charts, and the concept of function, in addition to traditional topics. A series of reports and papers on various topics contain many suggestions for putting these recommendations into effect, especially in grades seven through nine. Also included are an annotated bibliography on applications of mathematics, and a list of topics for the use of mathematical models. (MM)

This document is the report of a curriculum writing session, and covers three activities: (1) a textbook for grade seven, written using the outlines produced the previous year (SE 012 735), was reviewed and criticized; (2) the outlines for grades eight and nine, also produced the previous year, were completed in greater detail; and (3) considerable attention was given to possible sequences of courses for grades 10-12. Broad outlines for several courses at this level were written, including a deductive block; a year of vector geometry, linear algebra, and elementary functions for grade 10; a course in analytic geometry and algebra; and a course in elementary functions and calculus. Several papers treating specific topics are included in this report. (MM)

The "Television Polytechnic" program of 1966 - 1970 was a joint effort of the Polish Ministry of Higher Education and Polish Television to provide televised education for working students. The subjects taught were those normally included in the first two years of the higher technical schools--math, physics, geometry, chemistry, mechanics of materials, and electrotechnics. Each semester lectures were given by professors from the higher technical schools, and students were provided with texts and access to consultation centers. Assessment indicated that television lectures were helpful to regular students as a complement to their classroom experiences and were a useful means of providing basic instruction to those not enrolled in a higher school. Preparatory and first year television courses will remain as permanent elements in the Polish educational system and the recent implementation of a second channel will lead to an expansion of programs, including many aimed at post-graduate audiences. (PB)

In a previous paper (‘Philosophical Transactions,’ A, 1897, vol. 189, p. 137) we have drawn attention to the fact that the disturbance set up in a liquid by the impact of a rough sphere falling into it, differs in a very remarkable manner from that which follows the entry of a smooth sphere. In the present paper we describe further experiments, made with the object of ascertaining the reason of this difference, and give the conclusions reached. It appeared desirable, in the first place, to take instantaneous photographs of the disturbed liquid below the water-line. These were easily obtained by letting the splash take place in an approximately parallel-sided thin glass vessel (an inverted clock-shade) illuminated from behind. The liquid surface when undisturbed was about level with the middle of the camera-lens, which was focussed for the sphere when under water. The general arrangement of the optical apparatus will be suffi­ciently understood from the accompanying cut (fig. 1). The method of timing the illumination was that already described ( loc. cit. ).

Publisher Summary This chapter presents experiments on deformations of paper in stacks, in which sheets were conditioned at a fixed humidity and then printed with a screen line plate by means of a Vandercook proof press. The prints were stacked progressively during printing. The stack was exposed to another humidity for a time and then, a second color was printed upon the first one, from the same plate. From the interference of the two line patterns, it was possible to measure deformation of the paper. Every sheet was measured and the results for an offset paper, a conventional coated paper, and a machine-coated paper are presented in the chapter. The change in dimension is given on the abscissa in units of 0.1 mm/m. The trials were made on three papers: (1) an offset paper, (2) a conventional coated paper, and (3) a machine-coated paper. The results show that the deformation was greater in the transverse direction than in the longitudinal direction and the deformations were not the same whether the transverse direction or the longitudinal direction was parallel to the cylinder axis; this effect shows that the press has a laminating action.

In a previous paper (‘Philosophical Transactions,’ A, 1897, vol. 189, p. 137) we have drawn attention to the fact that the disturbance set up in a liquid by the impact of a rough sphere falling into it, differs in a very remarkable manner from that which follows the entry of a smooth sphere. In the present paper we describe further experiments, made with the object of ascertaining the reason of this difference, and give the conclusions reached. It appeared desirable, in the first place, to take instantaneous photographs of the disturbed liquid below the water-line. These were easily obtained by letting the splash take place in an approximately parallel-sided thin glass vessel (an inverted clock-shade) illuminated from behind. The liquid surface when undisturbed was about level with the middle of the camera-lens, which was focussed for the sphere when under water. The general arrangement of the optical apparatus will be suffi­ciently understood from the accompanying cut (fig. 1). The method of timing the illumination was that already described ( loc. cit. ).

The Supplement now offered to the Society, is intended to clear up certain points which have been represented to the author as obscure or doubtful in his former communications, and at the same time to connect more intimately the prismatic experiments of the second paper with those made upon convex glasses, and described in the author’s first paper on the subject. Since Dr. Herschel has heard the originality of his observation of the red bow called in question, upon the ground that a red bow had been observed by Sir Isaac Newton, which is merely the converse of the blue bow (the change of colour being dependent upon the direction in which the light is received upon the prism), Dr. Herschel first endeavours to answer the objection, and reminds us that in his former observations the angular breadth and elevation of the two bows are different; but those of the Newtonian blue and red bows are said to be, and are, necessarily equal. In the Newtonian experiment also, the same beam of light is made to exhibit both phenomena, being received upon two right-angled prisms, applied base to base, so that one portion of the light is reflected upwards, as a blue bow from the under surface of the first prism; and the remainder, by transmission, through the second prism, appears as a red bow to an eye beneath. But in Dr. Herschel’s experiment, the same prism is made to exhibit, to an eye in the same situation, the red bow as well as the blue, by means of light transmitted in an opposite direction through the under surface of the prism, without any occasion for a second prism, which (as Dr. Herschel observes) is necessary in the Newtonian method of conducting the experiment.

his test shows how the rigidity is worked out by applying those figures to the formulas which are calculated with Clark's Rigidity-tester, when several sorts of Washi are left in the atmosphere of the different temperatures and humidities and then balanced.The rigidity of Washi is less influenced by the temperature within the limits around the ordinary temperature ; however, it is influenced by the change of the atmospheric relative humidity.In general, the bigger becomes the humidity, the smaller becomes the rigidity.But, this change can hardly be recognized in case of machine-made foreign-style paper, while in cas of Nashio-paper, for instance, mixed with clay, humidity of the mixtures is considered to be greatly influential on.The phase of paper's textile-arrangements generally enables us to imagine that the vertical course differs from the horizontal course in rigidity ; so it is, in reality ; the horizontal rigidity is smaller than the vertical one.

With a view to finding a better method of separating the different components during electrophoresis on paper, we have carried out the electrophoresis in two dimensions, first flowing the current in one direction and then at right angles to that direction. For this purpose we used square paper sheets between two glass plates of the same dimensions. The electrical behavior of the material to be analyzed was tested before electrophoresis, so that it was possible in each instance to establish the duration and the different electrical characteristics of the two periods of electrophoresis and also the best initial position for placing the material. The experiments show that the components are found along a straight line, the position of which is determined in the plane by the characteristics and time flow of the current during the two phases of the process. Referring to the initial position of the material, each individual component is to be found at the extremity of the hypotenuse of a right angled triangle, the other sides of which are formed by the paths made by the substance in the two different phases of the process. The obvious advantage of the two-dimensional electrophoresis on paper is that it is possible to trace, on the same sheet of paper, the path taken by the different components under two different electrical conditions.

Contact angle was discussed, relation on P versus, √t on three species of the sanAple were plotted for five kind of oil, and the first slope (F. S.) and the second slope (S. S.) were determined.If ?A=O i. e. cos θ=1, the ratio of F.S. and S.S. between two kinds of oil in the same paper (Y axis), should have a straight line to the ratio √surface tention/viscosity coeffcient of each oil (X axis), and the line should pass the original point, its slope should be 1.The regression line on 70 point of the experimental value, clossed the Y axis at -0.101 and its slope was 1.106, and we have tested statistically that this line was not different significantly to the theoretical line.From the fact mentioned above and the fact that in some cases the slope on S.S.is greater than that on F.S., it is likely that the contact angle is a little larger than zero.Assuming that the mean hydraulic radius which was calculated from the data on the oil penetration method by applying Kozeny-Carman equation is equal to the average paper capillary radius, we can calculate number of capillary per unit area and the product of mean hydraulic radius and number of capillary. Hydraulic radius was measured with the constant pressure air permeability apparatus and number of capillary was calculated from these radii. But the physical meaning of number of capillary are not clearly understood.On the discussion from the various consideration, it might be concluded that paper should have a different layer on vertical direction. On the paper that the slope of F. S. is greater than that of S.S. penetrating velocity have to be determined on the second layer and so, relation between the number of capillary product with mean hydraulic radius and hydraulic radius by permeability method, should have good correlation value. Actually we have the correlation coefficient almost nearly. The correlation coefficient on the first layer should be lower than that on second layer, and actually it was 0.35.

Hmax and smoothness were measured by needle touch type unevenness tester and Bekk automatic smoothness tester, respectively, and relation between h and Hmax or smoothness was examined.It was found that h was closely conected with Hmax or smoothness.When thickness of paper was measured in superposition by micrometer, and h was obtained with the formula showed in part 1, thereby the surface condition of paper is briefly known.

Discussion of the problem of an elastic half-space with spherical cavity is continued in respect of Rayleigh waves on the plane boundary. Displacements in the initial and first group of higher order Rayleigh waves are derived by using the time-harmonic solution developed in part I of this series with attention confined to the case of time-harmonic normal stress at the cavity. These are employed to find also the response to an exponential shock at the cavity and graphs are presented showing the surface motion due to the initial Rayleigh waves. Finally, in an appendix to the paper, some comments are given on a recent paper by R. D. Gregory on the problem of the half-space with cavity.

A method is described for making a projection drawing from any point of view in which the drawing is set out directly from the dimensions of the object without the use of intermediate plans and elevations. Seven key figures are given with the necessary numerical factors which give directly the proportions for forty-two points of view. By the use of square ruled paper and these figures, aided by the slide rule, figures may be drawn in which the dimensions are found by means of the rulings on the paper only, so that scales of length are not needed. An example of the use of the method is an illustration of two pentagonal dodecahedrons, one that of Euclid and the other that of a crystal of iron pyrites. The last part of the paper deals with the solution of the spherical triangles on which the method depends.

er demonstrating case, I finally invented the form of folding paper display case shown in Figure 1. This case is easily made by using a No. 160 white card paper. The design is mapped out, cut and creased, as inFig. 2.—Plan of demonstrating case. A 160-pound white bristol cardboard is used. The design is mapped out according to the dimensions indicated. The cardboard is cut on all solid lines except slits marked A, B, C and D, also 1, 1, 1 and 2, 2. 2, which are to be done later. All dotted lines are creased with some blunt instrument, the cardboard being placed over soft wood. As many apertures as are desired are punched out, 1*4 inches for each being allowed. All creased lines are bent forward except a, b, c, d, e and /, which are to be bent backward, and are folded as in illustration. The flaps A, B and C, D should overlap; slits are cut in opposite direction, as indicated, so that they will properly lock. With a sharp instrument are cut small slits 1 and 2, which will go through three thicknesses of paper that make up the back of the case, and by which the whole is fastened with ordinary paper fasteners.

A CIRCULAR filter paper suitable for partition paper chromatography is cut so that it can be made into a cone. A number of origins are marked at adequate intervals near the lower edge, the standard and unknown solutions are dropped on these origins (see Fig. 1) and chromatographic development allowed to take place in an upward direction. The cone is immersed in the solvent in a Petri dish placed in a glass vessel with a tight lid (see Fig. 2).

Fluid flow and heat transfer in non-circular passages, with peripheral variation of heat flux around their boundaries, is of particular interest in the nuclear engineering field. This paper deals with heat transfer to air (Prandtl number ⋍ 0.7) in fully-developed turbulent flow in an annular passage, the outer boundary of which is insulated. Experiments have been conducted on annuii whose radius ratios are 1.632, 2.583, and 3.875 over a Reynolds number range 10 000–50 000. Pressure drop, velocity and temperature distributions in the cross-section have been measured over the whole range of eccentricity of the annulus geometry. The effect of eccentricity on average friction, average heat-transfer coefficient, and wall temperature distribution for uniform heat generation within the heated boundary has been studied. The results of the experimental programme are compared with the semi-theoretical predictions and experimental data of other workers in this field. For completeness, a theoretical equation for the heat-transfer coefficient in an internally heated concentric annulus has been derived.

Toward the end of last school year nearly every member of our plane geometry classes undertook, in addition to his regular class work, the preparation of a paper on any subject of interest to him. The subject, chosen without reference to mathematics. was investigated to see whether or not it was in any way dependent upon, or connected with, mathematics. This plan originated with the pupils of the classes—probably as a reaction to the class work, which was designed to stimulate thought along that line.

The work reported in the second and third papers revealed that under high speed operating conditions significant slide occurs at the contact points. The full extent of the present limitations became apparent in 1967 and led in that year to the new race design. In this design the balls contact the outer track at two points simultaneously instead of the conventional one point. An approximate analysis of race behaviour is presented which indicates that when the new bearing is driven from the inner track the ball motion is controlled at this track over most of the operating range. This fact and the retention of the inner, and the major outer contact positions, at points close to their static positions result in improvements in performance of up to 30 per cent at high speeds.

(1928). LXXII. The stream-line motion of fluid in a curved pipe (Second paper) The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science: Vol. 5, No. 30, pp. 673-695.

A theory of psychophysics is discussed that enlarges the classical theory in three general ways: (1) the multidimensional nature of perception is made explicit; (2) the transformations of the theory are interpreted geometrically; and (3) attributes are distinguished from sensations and only partially ordered. It is shown that, with the enlarged theory and some elementary geometry, one may use a few ideas to explain many qualitative results, to give a credible and precise account of quantitative properties of data, and to design experiments which seem worth doing whether the theory is valid or not. In developing the new theory, the major criticisms of the classical theory are answered. This paper is limited to one of three aspects of the typical psychophysical measurement experiment--the relation between perceptions and attributes. Although the theory has been extended to category and rating scales, only magnitude estimation and related measurement procedures are discussed here. The scope of this paper is further limited to phenomena which have analogues in several sense modalities. (Author/KM)

This paper presents a method of encoding geometric line-drawings in a way which allows sets of such drawings to be interpreted as formal languages. A characterization of certain geometric predicates in terms of their properties as languages is obtained, and techniques usually associated with generative grammars and formal automata are then applied to the geometric framework. Section 1 of the paper specifies the geometric framework; section 2 develops the background material on formal languages, grammars, and automata; section 3 is concerned with geometric predicates; section 4 covers invariance theorems; and section 5 indicates areas for further investigations. (DT)

This NSF-sponsored conference on the teaching of mathematics at the junior high level involved mathematics teachers and teachers of the natural and social sciences. Papers written for the conference form the bulk of this report. Summaries of the papers and general discussions are organized into shorter reports to give some guidelines which could be useful to any group wishing to create curricular materials for junior high school mathematics or simply to get some ideas for direct application to the classroom. The topics of the report were: what mathematics junior high students do and do not know; new emphases in content; mathematics in geography, social science and biology; teaching strategies and styles; mathematics and language; and teaching training. (JP)

These materials were written with the aim of reflecting the thinking of Cambridge Conference on School Mathematics (CCSM) regarding the goals and objectives for school mathematics K-6. In view of the experiences of other curriculum groups and of the general discussions since 1963, the present report initiates the next step in evolving the "Goals". Three areas considered in this report are geometry, functions in preparation for calculus, and applications. Two working papers are presented on applications - probability and mechanics and slopes. One working paper on circular functions is included. Fifteen working papers are presented involving geometry and geometrical concepts. The papers on geometry include examination and description of common objectives; playing with figures, blocks, and tessellations; constructions; graphs and polygons; tessellations; dissection of figures; order; measurement; similarity and map making; symmetry, congruence, and rigid motion; transformation groups; rotations and matrices; iterated reflections in mirrors; knots; and spheres, cylinders, and torus. [Not available in hard copy due to marginal legibility of original document]. (RP)

The author describes a way of teaching a one-year geometry course which includes plane geometry, solid geometry, coordinate methods of proof, and vector methods of proof. His method is based upon encouraging students to use the properties of geometric figures that they already accept without formal proof. He employs paper folding extensively to arrive at many of the plane geometric concepts. (CT)

THE FIRST PAPER, "REPORT OF MATHEMATICS CONFERENCE," IS A SUMMARY OF DISCUSSIONS BY 29 PARTICIPANTS IN A CONFERENCE ON CURRENT PROBLEMS IN MATHEMATICS EDUCATION RESEARCH. REPORTED ARE (1) RECENT PROGRESS, PROBLEMS, AND PLANS OF CURRICULUM DEVELOPMENT GROUPS, (2) GENERAL FORMULATION OF CURRICULUM AND METHODS, (3) TEACHER TRAINING, (4) COMMUNICATION, AND (5) RESEARCH MANPOWER. THE SECOND PAPER, "ELEMENTARY MODERN MATHEMATICS FROM THE ADVANCED STANDPOINT," GIVES A RECOMMENDED SYLLABUS FOR THE FIRST FIVE SEMESTERS OF A SIX-SEMESTER COLLEGE MATHEMATICS COURSE TO BE TAKEN BY POTENTIAL HIGH SCHOOL TEACHERS. FINALLY, THE THIRD PAPER, "SCHOOL MATHEMATICS STUDY GROUP (SMSG) AND THE GIFTED CHILD," REPORTS THE REASON FOR INCLUDING EXPLICIT RECOMMENDATIONS FOR EDUCATING THE GIFTED CHILD IN ANY LONG-RANGE CURRICULUM REFORM. (RP)

A stress‐strain analysis of single cubic crystals is developed which utilizes the strain data supplied by the x‐ray back‐reflection divergent beam method. The principal strains and their directions are determined and from the principal strains and the known elastic constants the complete stress‐strain configuration is obtained. Thus the maximum magnitude and the direction of the shearing strain on a given set of crystallographic planes are obtained and the set of planes on which the maximum value of the shearing maxima occurs is also determined. From a knowledge of the stress‐strain configuration, the stored elastic energy of the crystal is deduced; it can be partitioned into two components, that due to shearing strains and that due to a mixture of normal and shearing strains.The conditions under which the principal stress system coincides with the principal strain system are also investigated. Furthermore, a number is constructed that measures the distortion of the crystal in terms of the energy incremen...