bluuebunny/crossref_metadata_2025 huggingface dataset:
Increased expression of interleukin-1 receptor type 1 in active endometriotic lesions
Authors: Lawson C , Al-Akoum M , Maheux R , Akoum A
Published: January 2007

The establishment and progression of ectopic endometrial implants are dependent upon their interaction with and responsiveness to the stimuli present in their new environment. According to our and other previous studies, immune cells-derived cytokines, such as IL-1, may alone or in concert with estrogens, enhance the capability of ectopic endometrial cells to implant and develop into the host tissue. In the present study, immunohistochemical and dual immunofluorescence analyses showed that the functional signaling interleukin-1 receptor type 1 (IL-1RI) is expressed in endometriotic tissue, particularly in the glands, and identified endothelial cells, macrophages, and T-lymphocytes as cells having marked expression of IL-1RI. The highest concentrations of IL-1RI protein in endometriotic tissue, as evaluated using histological score (HSCORE) and measured by ELISA, were found in red endometriotic lesions as compared with typical black–blue or white lesions. Western blotting showed a significant increase in the levels of the 50 kDa band, whose apparent molecular weight corresponds to the soluble form of IL-1RI. RT-PCR analysis of IL-1 mRNA levels showed a pattern of expression comparable to that of the protein. Interestingly, IL-1RI expression was more significant in the proliferative than in the secretory phase of the menstrual cycle. Marked expression of IL-1RI, the functional signaling receptor that mediates cell activation by IL-1, in red endometriotic implants, which are highly vascularized and represent the earliest and most active forms of the disease, point to a higher cell receptivity for IL-1 in these lesions, a relationship with the activity of the disease and a possible involvement in the early steps of endometriotic tissue growth and development.
Functional genomics studies of oocyte competence: evidence that reduced transcript abundance for follistatin is associated with poor developmental competence of bovine oocytes
Authors: Patel Osman V , Bettegowda Anilkumar , Ireland James J , Coussens Paul M , Lonergan Patrick , Smith George W
Published: January 2007

Poor oocyte competence contributes to infertility in humans and livestock species. The molecular characteristics of such oocytes are generally unknown. Objectives of the present studies were to identify differences in RNA transcript abundance in oocytes and early embryos associated with reduced oocyte competence and development to the blastocyst stage. Microarray experiments were conducted using RNA isolated from germinal vesicle stage oocytes collected from adult versus prepubertal animals (model of poor oocyte competence). A total of 193 genes displaying greater mRNA abundance in adult oocytes and 223 genes displaying greater mRNA abundance in prepubertal oocytes were detected. Subsequent gene ontology analysis of microarray data revealed significant overrepresentation of transcripts encoding for genes in hormone secretion classification within adult oocytes and such genes were selected for further analysis. Real-time PCR experiments revealed greater abundance of mRNA for βA and βB subunits of inhibin/activin and follistatin, but not the α subunit in germinal vesicle stage oocytes collected from adult versus prepubertal animals. Cumulus cell follistatin and βB subunit mRNA abundance were similar in samples collected from prepubertal versus adult animals. A positive association between time of first cleavage (oocyte competence) and follistatin mRNA abundance was noted. Follistatin, βB, and α subunit mRNAs were temporally regulated during early bovine embryogenesis and peaked at the 16-cell stage. Collectively, results demonstrate a positive association of follistatin mRNA abundance with oocyte competence in two distinct models and dynamic regulation of follistatin, βB, and α subunit mRNAs in early embryos after initiation of transcription from the embryonic genome.
Rabbit somatic cell cloning: effects of donor cell type, histone acetylation status and chimeric embryo complementation
Authors: Yang Feikun , Hao Ru , Kessler Barbara , Brem Gottfried , Wolf Eckhard , Zakhartchenko Valeri
Published: January 2007

The epigenetic status of a donor nucleus has an important effect on the developmental potential of embryos produced by somatic cell nuclear transfer (SCNT). In this study, we transferred cultured rabbit cumulus cells (RCC) and fetal fibroblasts (RFF) from genetically marked rabbits (Alicia/Basilea) into metaphase II oocytes and analyzed the levels of histone H3-lysine 9-lysine 14 acetylation (acH3K9/14) in donor cells and cloned embryos. We also assessed the correlation between the histone acetylation status of donor cells and cloned embryos and their developmental potential. To test whether alteration of the histone acetylation status affects development of cloned embryos, we treated donor cells with sodium butyrate (NaBu), a histone deacetylase inhibitor. Further, we tried to improve cloning efficiency by chimeric complementation of cloned embryos with blastomeres from in vivo fertilized or parthenogenetic embryos. The levels of acH3K9/14 were higher in RCCs than in RFFs (P<0.05). Although the type of donor cells did not affect development to blastocyst, after transfer into recipients, RCC cloned embryos induced a higher initial pregnancy rate as compared to RFF cloned embryos (40 vs 20%). However, almost all pregnancies with either type of cloned embryos were lost by the middle of gestation and only one fully developed, live RCC-derived rabbit was obtained. Treatment of RFFs with NaBu significantly increased the level of acH3K9/14 and the proportion of nuclear transfer embryos developing to blastocyst (49 vs 33% with non-treated RFF, P<0.05). The distribution of acH3K9/14 in either group of cloned embryos did not resemble that in in vivo fertilized embryos suggesting that reprogramming of this epigenetic mark is aberrant in cloned rabbit embryos and cannot be corrected by treatment of donor cells with NaBu. Aggregation of embryos cloned from NaBu-treated RFFs with blastomeres from in vivo derived embryos improved development to blastocyst, but no cloned offspring were obtained. Two live cloned rabbits were produced from this donor cell type only after aggregation of cloned embryos with a parthenogenetic blastomere. Our study demonstrates that the levels of histone acetylation in donor cells and cloned embryos correlate with their developmental potential and may be a useful epigenetic mark to predict efficiency of SCNT in rabbits.
Genetic Mapping of Biomass Production in Tetraploid Alfalfa
Authors: Robins Joseph G. , Luth Diane , Campbell T. Austin , Bauchan Gary R. , He Chunlin , Viands Donald R. , Hansen Julie L. , Brummer E. Charles
Published: January 2007

Biomass production represents a fundamental biological process of both ecological and agricultural significance. The genetic basis of biomass production is unknown but asssumed to be complex. We developed a full sib, F1 mapping population of autotetraploid Medicago sativa (alfalfa) derived from an intersubspecific cross that was known to produce heterosis for biomass production. We evaluated the population for biomass production over several years at three locations (Ames, IA, Nashua, IA, and Ithaca, NY) and concurrently developed a genetic linkage map using restriction fragment length polymorphism (RFLP) and simple sequence repeat (SSR) molecular markers. Transgressive segregants, many of which exhibited high levels of heterosis, were identified in each environment. Despite the complexities of mapping within autotetraploid populations, single‐marker analysis of variance identified 41 marker alleles, many on linkage groups 5 and 7, associated with biomass production in at least one of the sampling periods. Seven alleles were associated with biomass production in more than one of the sampling periods. Favorable alleles were contributed by both parents, one of which is from the M. sativa subsp. falcata germplasm. Thus, increased biomass production alleles can be gleaned from unadapted germplasm. Further, the positive quantitative trait locus (QTL) alleles from the parents are partially complementary, suggesting these loci may play a role in biomass production heterosis.
Acting Is Reacting
Authors: Mullin Dan W.
Published: January 2007

Acting has been variously defined as impersonating, imitating, interpreting, creating, feeling, radiating, being, behaving, or believing. Acting has even been defined as acting. In his book, The Actor's Ways and Means, Michael Redgrave states:

The essence of acting… is the power to act. Thought or emotion may or may not be present—but the basic will of the actor must be, quite simply, to act: not to think, not to feel, not to exhibitionize, not to make some personal statement—though he may do one or all of these—but to act.

Established actors, needing no definition, may accept any misty generalization if they are not sure what it is they are doing. The student actor needs a definition to guide him in understanding the process at any specific moment in the performance. The least common denominator in the actor's performance is a single unit. This unit is compounded of a stimulus and a response.
Public and Private Problems in Modern Drama
Authors: Peacock Ronald
Published: January 2007

It has become common to view drama in the post-Ibsen period as falling into two broad categories. On the one hand there was a strong and persistent tradition of the A Doll's House type of play, called for convenience “social problem plays,” and on the other a number of diverse styles of drama that represent counter-realism; plays in verse, expressionism, formalistic styles as in Yeats’ plays, revivals of myths, fantastic drama, surrealism, plays of Freudian psychology, Cocteau-ish poésie de théâtre, and so on, all of which, however different from each other, have in common that they turn away both from social problems and from the dramatic style associated with them. They do not necessarily, however, renounce realism for “romance,” or for something “poetic” in the escapist sense. Neither are the themes they treat always without relevance to the social situation.
II Tragico Imperatore
Authors: Bentley Eric
Published: January 2007

A young man loves a woman and is not loved in return. What is more, he has a rival. In a costumed cavalcade, the rival causes the young man's horse to slip, and the young man falls, faints, and, when he comes to, is the victim of the delusion that he is the person whose costume he is wearing: the German Emperor Henry IV. His sister converts a villa into a replica of this Emperor's palace so that the young man can live on as Henry IV undisturbed. After twelve years, however, the delusion wears off. Our man, no longer so young, decides not to let anyone know it and to stay on as Emperor, though sane.

After eight more years, his sister dies. But she had visited him shortly before her death and gained the impression that he might now be curable. She tells her nephew this, and soon after her death he brings a psychiatrist to the villa to see what can be done.
On Some Early Propositions of Statics
Authors: Bryan W E.
Published: January 2007

The current method of founding the teaching of Theoretical Statics on the experimental proof of the “Parallelogram of Forces” seems open to grave objection. The experiment is too remote from the facts of ordinary observation; and it is not so very easily performed. Moreover, the result, when accurately obtained, appears to the pupil to be so remarkable, that it is a pity not to reserve it for use as a verification of the soundness of pure reasoning.
Japanese Mathematics
Authors: W. J. G.
Published: January 2007

The Jahresbericht der Deutschen Mathematiker-Vereinigung has lately published an address given by Professor Harzer on “The Exact Sciences in Old Japan.” At first sight it seems difficult to see why the celebration of the birthday of the Emperor of the Germans should have been selected for the delivery of an oration on this topic. The Professor is himself an astronomer, so that it is interesting to note that he brushes aside as of little value anything that Japan has accomplished in the pursuit of his favourite science. Nor does he discover any ground for supposing that anything will be accomplished along this line by those whose sudden prominence in other realms has given the Western world so furiously to think. The history of the science of mathematics in the Island Empire is full of paradoxes. In the days of Pythagoras, it will be remembered, the secrets of the science were entrusted to what was to all intents and purposes a Masonic body, bound by oath not to divulge the mysteries in which they participated. And this body had a further curious and unexpected resemblance to the Masonic order as seen among the Latin races, inasmuch as its intrusion into politics led to more than one popular outbreak. In Japan the parallel almost holds good, save for the utilisation of the society for political purposes. The elementary parts of the science were open to all. He who ran might read. But before the student was permitted to penetrate into the arcana he was compelled to join a brotherhood and to take an oath of secrecy. Things have, of course, considerably changed since those days. The powers that be have become more and more convinced of the paramount necessity of improvement in the direction of mathematical training. And with the courage and directness which is such a marked characteristic of this astonishing race, they refuse to allow entrance to any of the higher courses at the University or higher technical institutions to those who have not the necessary preliminary mathematical equipment. We shall be very much mistaken if the results of these drastic measures do not make themselves very evident in the course of the next half-century.
Asymptotes
Authors: Nunn T. P.
Published: January 2007

According to the standard-text-books, an asymptote is a line which meets a curve at two points at infinity, but is not wholly at infinity. Regarded as a flash of mathematical wit the definition is clever and illuminating, but it is far too subtle for a raw student. He may have learnt that to call parallel lines “lines that meet in a point at infinity” is merely a Pickwickian way of saying that they do not meet at all, but, being in a plane, had (so to speak) their chance of meeting and just failed to take it; but even if he understands that, he may still be mystified here. For the asymptote and the curve meet, he is told, at two points at infinity, and who could guess what complicated tragedy of failure that euphemism conceals?
An Introduction to Exponentials for Technical Students
Authors: Bickley W. G.
Published: January 2007

The method of introducing e so common in text-books based upon an algebraic proof of the exponential series may be satisfactory for mathematical students, but it does not meet the needs of technical students (especially evening students), for it is impossible, if even desirable, to treat the matter along these lines in a manner at once brief, exact, and comprehensible. Moreover, it suffers from the great disadvantage that the property of e, which is of prime importance in technical applications, namely that ex is the function appropriate to the expression of the law of growth or decay so common in these—in other words, that ex is the solution of the differential equation dy/dx=y —appears as an afterthought, and not as the fundamentally important property, the raison d’être of the inclusion of e in the course and in the engineering formulae. The method outlined below, which has been tried for some few sessions, aims at presenting the differential equation as arising in the attempt to solve a concrete practical problem, and the discovery and identification of two solutions—one a series, the other in terms of indices. By obtaining the series first some idea of the use of non-terminating series, and appreciation of convergence, are gained. To simplify the numerical calculations, the index solution is then found. Here e appears as a limit, and to obtain its numerical value recourse must be had to the series.
Blaise Pascal
Authors: Milne J.J.
Published: January 2007

Most mathematical students are familiar with Pascal’s statement, made in a communication to a learned society in Paris in 1653, that he had a number of mathematical treatises, chiefly geometrical, ready for publication, amongst them being the following : Conicorum opus completum, et conica Apollonii et alia innumera unica fere propositione amplectens ; quod quidem nondum sex decimum aetatis annum assecutus excogitavi ; et deinde in ordinem congessi.
A Simple Theory of the Gamma-Function
Authors: Eagle Albebt
Published: January 2007

Introduction. Many students interest themselves in finding a function which is equal to factorial n for integral values of n. There is a certain fascination about this function : it is so different from the transcendental functions of applied mathematics.
The Anharmonic Ratio of Four Points in Space and the Discord of Two Pairs of Points
Authors: Muirhead R. F.
Published: January 2007

§ 1. The Anharmonic Ratio expressed in terms of Squared Distances. In Cremona’s notation, the symbol (AB, CD) stands for the quantity

where ABCD are four collinear points, A, B and C, D are Conjugate pairs. It is easy to show that
Anharmonic Coordinates
Authors: Coddington F.J.O.
Published: January 2007

Hamilton, in his Elements of Quaternions (Book I. Chapter II.), developed this system of coordinates, by means of his vectorial algebra, in connection with his theory of geometrical nets.

In the following paper I have re-proved the fundamental theorems, by other methods, and have sketched out the natural lines of development of the system.

Though few, if any, of the results arrived at are new, they are at any rate newly stated; and I believe that the elegance of method will command interest.
Note on the Teaching of Analytical Geometries
Authors: Robson A.
Published: January 2007

In his very interesting article on Asymptotes in the May Gazette, Prof. Nunn points out that the mathematician often asserts that a line always meets a conic in two points which may be at infinity, and that teachers and text-books dwell too little upon the fact that the word “point” is used in widely different senses.
On the Geometrical Method
Authors: Larmor J.
Published: January 2007

As I can pretend to only limited practical experience in the matter of teaching the foundations of Mathematics, and of Geometry in particular, I have thought that the time allowed for my remarks may be advantageously occupied in a rapid survey of what the geometrical method has accomplished, and the estimation in which it has been held at different periods of the history of the development of Mathematical Science. Thus, avoiding expert questions relating to Education, we may begin by considering from a general standpoint what is implied in the science of Geometry, for the right ordering of which, in its educational relations, this Association has been working for a long series of years. It would, I presume, be against the instincts of our members to interpret the subject in any narrow sense. Geometry is usually based on the partly mathematical, partly logical, framework which has descended to us from Greek times, and which is chiefly associated with its earliest great systematization in the work of Euclid. But modern development would perhaps render any definition of its scope incomplete that did not include all methods in formal and physical Mathematics which proceed by direct contemplation of the relations of the subjects considered, instead of through representations of them that are merely quantitative or numerical.
Four Fours. Some Arithmetical Puzzles
Authors: Rouse Ball W. W
Published: January 2007

An arithmatical amusement, said to have been first propounded in 1881, is the expression in the expression in the ordinary arithmatic and algebric notation of the consecutive integers from 1 upwards, as far as practicable, by the use of four “ 4 ”. I have mentioned the problem in my Mathematical Recreations, but my friend Mr. Oscar Eckenstein has now carried the solutions considerly further: I think a bare statement of our procedure may be of interest, and perhaps some readers ot the Gazette may amuse themselves by filling in the details ot the analysis or applying similar methods to higher numbers.
Proof by Recurrence
Authors: Lupton Sydney
Published: January 2007

In its most simple and general form the proof by recurrence of a formula containing constants and one variable consists in giving a number of successive integral values to the variable, and showing by trial that the formula is true for those values. The next step is to show that the form of the result remains the same when n+1 is substituted for n, or that each result depends upon the preceding one. If this second step is neglected, as it often used to be, the proof seems to be invalid.

This proof by mathematical or successive induction has been very generally accepted by mathematicians and neglected by logicians.
On Tertial, Quintal, Etc., Fractions
Authors: Cunningham Lt.-Col. Allan
Published: January 2007

1. Def. The reciprocal (1/N) of any number (N) will be termed a Binal, Tertial, Quintal, ... Decimal, ... r-mal fraction, when expressed in the scale whose radix is r = 2, 3, 4, ... 10, ... r respectively. In this paper the developments will be given for the most part in a general form in the r-scale; the examples will be in the scales of r = 3 and 5. The notation is similar to that of ordinary decimals. Thus (see Tables at end):
Approximation to , where n is an Integer and 0<x<1.: (Concluded.)
Authors: Child J. M.
Published: January 2007

Convergents to tan nθ.

If the numerator and denominator of the fraction are both convergent series, one value of tan nθ is
The Beginnings of Arithmetic
Authors: Filon L. N. G.
Published: January 2007

1. It is a commonplace of Biological Science that each individual, in his physical development, reproduces in an abridged form the successive stages through which the race has passed in its remote history. Educational experience suggests that a similar process occurs in mental development, so that, generally speaking, ideas have to be introduced in the order of their historical occurrence, if unnecessary resistance is not to be encountered.
The Teaching of Geometry
Authors: Hamilton J. G.
Published: January 2007

In the teaching of mathematics as of other subjects the primary determining factor ought to be, but seldom has been, the consideration of the pupil himself—his interests, aptitudes, and capabilities. The teacher should be free to devise the best means of securing his fullest co-operation, and examinations should allow the greatest possible elasticity. It is the reversal of the rational order that has wrought so much harm in the past and that so long delayed and still impedes reform.
On the Axioms and Postulates Employed in the Elementary Plane Constructions
Authors: Macaulay F. S.
Published: January 2007

By the elementary plane constructions we mean the bisection of angles and segments of lines and the drawing of perpendiculars. The constructions given below are extremely simple, and it is very possible that some or all of them have been given before; but they seem to be of sufficient interest to be reproduced, for the reason that, if they are not new, they are not generally known.
A Suggested Rearrangement of the Book-Work on Some Elementary Series
Authors: Bromwich T. J. I’a
Published: January 2007

In all the English text-books on Analytical Trigonometry, so far as I know, the power-series for cos x; is obtained as the limit, when n tends towards infinity, of the finite sum for cos x,

where and the number of terms is equal to the integer next greater than . The power-series for sin x is obtained similarly; the two series and the method of finding them being both due to Euler.
A Note on Tucker’s Harmonic Quadrilateral
Authors: Langley Edwakd M.
Published: January 2007

In a previous note it has been pointed out as a corollary to the proof of Ptolemy’s Theorem that, if in a cyclic quadrilateral ABCD, AB.CD = BC.DA, and if M be the mid-point of the diagonal BD (2m),

and

so that A and C are each the image in DB of the inverse of the other w.r.t. M, the constant of inversion being m2.
The Finger Method of Multiplication and Extensions of It
Authors: Lodge A.
Published: January 2007

We are greatly indebted to the correspondents who have supplied so much valuable information regarding the history and development of the finger-method of multiplication, and to our Editor for the summary of them which will be found in the notes in this number of the Gazette.
A Problem in Fourfold Geometry
Authors: Mair D. B.
Published: January 2007

We take two bodies P Q which have straight world lines OP KQ. This means in classical language that P and Q have a constant relative velocity. Our object is to find the configuration which an observer U reckons that of nearest approach of P and Q, and to find U’s measure of the least distance.
The Graphical Treatment of Power-Series
Authors: Milne William P.
Published: January 2007

The object of the “Advanced Section” being to study and find out the best methods of teaching Scholarship work, it seems advisable briefly to pass in review the higher algebra syllabus which is at present being taught to pupils in the departments of advanced mathematics in our secondary schools.
The Asymptotes of Plane Cueves
Authors: Green H.G.
Published: January 2007

While Professor Nunn’s treatment of Asymptotes (Math. Gazette, May 1926) gives the elementary student a method for finding the equations of linear asymptotes, the present writer feels that there is some weakness in the method, as the geometrical consequences of the various types of asymptote on the general form of a curve are not emphasised. There is, moreover, a serious difficulty for the advanced student. Professor Nunn regards two parallel asymptotes as a degenerate case of the asymptotic parabola, thus throwing over entirely the ideas of multiple points at “infinity”. This must lead to considerable modifications in the values and properties of Pliicker’s numbers. In fact, the change would affect the whole theory of the deficiency of curves.
The Nature of Elementary Mathematics from the Modern Standpoint
Authors: Elliott C.
Published: January 2007

The character of the modern treatment of elementary mathematics is such that mathematics might reasonably be described as the Science of Classification. The mathematician may be supposed to begin as a collector, whether of butterflies, minerals, or even of cigarette cards, does not matter. Let us consider that he has collected so many different kinds of things as to become virtually a collector of collections, each collection having its own classification, and that his next step is to compare these classifications with each •other, that is, to become a classifier of classifications. The mere comparison of classes requires the i-iea of cardinal number, since the only thing which can be said about two classes, irrespective of the nature of their individual members, is the number of individuals in each class. But the comparison of classifications which do not consist merely of single classes introduces ideas which are essential to the further development of the idea of number, and to the development of Geometry, and which should therefore precede that development. It is in this super-classification or comparative study of classifications that the foundations of mathematics are to be sought, and the question therefore arises, whether the presentation of the rudiments of mathematics should not be modified so as to recognize this fact. Of the fundamental ideas of mathematics, four especially, namely those of correspondence, order, group, and of multiple correspondence, seem to lend themselves, in their elementary treatment at any rate, to the method in which they are regarded as belonging to the Science of Classification, and will be considered in that light here, the last three being regarded as developments of the first.
On the Representation of Imaginary Points by Real Points on a Plane
Authors: Lodge A.
Published: January 2007

I wish to express my great indebtedness to Mr. P. J. Heawood, of Durham University, for his careful and suggestive criticism of the contents of this paper, without, however, making him responsible for any statement in it with which he may disagree.-A. LODGE.

For the sake of those readers who have not seen the first part of this paper, and to whom the whole subject is new, I may state that the whole argument is based on the fact that any imaginary point such as (6 + i, 4 + 3i) has a simple vector x + iy = (6 + i) + i(4 + 3i) = 3 + 5i, and therefore may be vectorially plotted on to, or represented by, the real point [in this case (3, 5)] which has the same vector. This real point is thus the representative of all the imaginary points which have the same vector. The paper deals with the consequent relations between the representatives of conjugate imaginaries, especially when the imaginaries satisfy the equation of some given curve : in this case the two real points representing such a pair of conjugate imaginaries are called images of each other in the curve.
The Principle of Similarity
Authors: Dobbs W. J.
Published: January 2007

As teachers of elementary mathematics I suppose we are all familiar with the Circular 711 of the Board of Education. That circular is now out of print, and its place has been taken by Circular 851, which, I think, is not so well known. The purpose of the latter circular is to state the matter afresh, to explain it more fully, and to justify the attitude adopted. The main practical suggestion of these two circulars has been so largely adopted in present-day teaching in this country that I propose at the outset to base my remarks largely upon it.
Theorems Connected with Inversion
Authors: M‘Vicker C. E.
Published: January 2007

The square of the tangent from a point P to a circle S, divided by the diameter of S, will, in what follows, be denoted by (PS).

If S open out into a straight line, it is easily seen that (PS) equals in the limit the perpendicular distance of P from this line; accordingly the length of the perpendicular from a point P on a straight line S will be denoted by the same symbol (PS).
Missing-Figure Problems
Authors: Berwick W. E. H.
Published: January 2007

In a recent number of the Mathematical Gazette (March 1920) I proposed the following problem for solution:

Some of the figures in the long division sum have been replaced by crosses, and it is required to restore them. Four solutions are possible—of course in scale ten. It is understood that the sum is to be worked as in elementary arithmetic and that there is nothing fictitious. There must be no line of zeros, neither must the left-hand figure of any line be zero. In the divisor the digits figure ought not to be zero (for such a zero would ordinarily be cancelled before working the sum), but it is not enough to assume this: proof must be possible from the data. In the Four Fours the proof is instantaneous, because there is no remainder and the right-hand figure of the dividend is 4.
The Dissection of Rectilineal Figures
Authors: Macaulay W H.
Published: January 2007

I am not acquainted with any investigation of the problem of dissecting two given rectilineal figures of equal area, by straight lines, so that the parts of either will fit on the other, and the results given here are new to me. The problem is one which suggests a variety of interesting geometrical constructions. It is natural to consider first how, and in how many different ways, a dissection can be made so as to give the smallest possible number of parts ; but types of dissection which do not give the smallest possible number of parts may also be worth attention. I exclude cases of dissections depending on some accidental or unspecified coincidence of dimensions in the given figures.
The Theory of the Polar Planimeter: An Application of the Method of Barriers
Authors: Tomkins J. A.
Published: January 2007

Many instruments have been devised for mechanically recording the area of a closed curve when a tracing point is carried round its boundary, and, of these, probably the best known is the Amsler Planimeter invented in 1854 by Professor J. Amsler of Schaffhausen.

It belongs to the class known as polar planimeters, so called because the tracing arm turns about a fixed centre or pole.

The theory of the instrument has been given in various ways by different writers, but perhaps the neatest explanation is that given by, Professor Henrici in the British Association Report, 1894. As this is probably pretty well known, only the briefest resumé will be given here in order to make the extension of it and the present article readily intelligible. The explanation depends on the general theorem that the area, reckoned algebraically, swept out by a line of fixed length in a complete cycle, is equal to the difference of the areas described by its ends. In the Amsler Planimeter, one end of the tracing arm moves over the circumference of the circle described by the end of the pole arm while the tracing point at the other end describes the boundary of the area.
The Graphical Treatment of Differential Equations
Authors: Brodetsky S.
Published: January 2007

5. We shall now apply this method to a few typical cases.

(i) The equation y1=x2+y2

gives y2=2x+2y2(x2+y2).

Hence, y1 is always positive; y2 vanishes along the curve

r2=-cot θ,

in which x=r cos θ, y = r sin θ. Hence, in Fig. 5(a), the curves have the form) above the dashed curve (the locus of inflexion), and ( below this curve.
On the Teaching of Mathematics in Elementary Schools, and its Continuity with the Teaching in Secondary Schools
Authors: Jones R. W.
Published: January 2007

Arithmetic has been until recently the only branch of Mathematics generally taught in the Primary School; and as this subject deals exclusively with numbers in their simplest and most practical form, it must, in the order of teaching, be the first introduced to the child. The premier place must also be awarded to it when considering the relative importance of the various branches composing the unity of Mathematics.
On the Representation of Circles by means of Points in Space of Three Dimensions
Authors: Pedoe D.
Published: January 2007

The idea of the representation of a linear system of plane curves by means of a flat space is of fundamental importance in algebraic geometry. In this paper some details of the representation of circles are worked out, and an attempt is made to give a three-dimensional picture of certain parts of circle geometry The theorems given in italics are thought to be new, although in a subject with such an enormous literature it might almost be said that no theorem can be a new one.
The Statistical Theory of Turbulent Motion
Authors: Trubridge G. F. P.
Published: January 2007

The study of the turbulent motion of fluids is of great importance in aerodynamics and engineering, besides being a fundamental problem of hydrodynamics. Several attempts have been made to develop an adequate theory of turbulent motion, but, except for a few good empirical formulae, little advance has been made. L. Prandtl has given a clear and concise account of his own theory and that of Th. von Kármán in a recent paper (1), which contains full references to the literature of the subject. H. Bateman has also contributed an excellent account of the present state of the theory in the Report on Hydrodynamics of the National Research Council of America. In neither of these accounts, however, has reference been made to the theory put forward by J. M. Burgers of Delft, in a series of papers (2) published in 1929, and further developed in 1933. In these papers Burgers has attempted to apply the methods of classical statistical mechanics to the theory of turbulent fluid motion. In this article a simplified account of his methods will be given.
Similar Figures
Authors: van der Meersch A.
Published: January 2007

If it was required to divide the area in Fig. 1 into n equal parts—say, for simplicity, three equal parts—how should we proceed?

There is a method, given in books on geometry, of constructing a figure similar to a given figure and n times its area. This method pivots on a property of similar figures as follows: if ABCDE and

FGHKL are similar figures on bases AE and FL respectively, then

A1: A2 = AE2: FL2

For purposes of comparison with what follows, the solution by this method is here recalled.
What is Wrong with the Teaching of Approximations?
Authors: Inman S.
Published: January 2007

The reports of the London University on the General School Examinations throw an interesting light on the knowledge of candidates of the subject of approximations. The 1928-30 reports make it quite clear that the subject is badly taught and that glaring errors occur very frequently The 1929 report states that this fault is general and is not confined to particular schools. When it is considered that the University examines annually about 25,000 pupils from schools all over the country there is justification for the suggestion in the title of this paper.
Problems of Individual Education, with Special Reference to Work in Mathematics
Authors: Spriggs G. W.
Published: January 2007

AT the outset I desire to express my deep appreciation of the honour of being invited to lay before your Association some of the investigations into educational practice which I have undertaken during the past few years. I ask only that they be regarded aa any scientific experiment, as a search for some true principles, and that at any stage of that quest failure was to be welcomed as warmly as success, as affording suggestions for further enquiries. That the experiment has been attended with a measure of success does not make me ready to proclaim it as the ne plus ultra : I submit the facts to your judgment in the hope that they may afford material for a profitable discussion and stimulate some abler investigators to pursue the matter further.
The Mathematician in Ordinary Intercourse
Authors: Hudson H. P.
Published: January 2007

I received this morning a pamphlet in which the master of Stowe School, describing the Acropolis, says: “The physical beauty of the place alone would have moved even a mathematician’s heart,” and that appears to be what the world thinks of us, or, at any rate, a different part of the scholastic world. I am, of course, speaking to mathematicians and not to teachers, and I suppose we have all at one time or another suffered from the misguided hostess who introduces one as a mathematician. When that terrible word is mentioned the other person instantly curls up and says to the so-called mathematician, “How terribly clever you must be,” or else, “I never could do maths, at school.” Neither is a good opening in conversation, and it takes some time to beat down that barrier and get back to ordinary friendly terms. A mathematical reputation is rather like a clerical collar; it is a hindrance to intercourse and condemns one to a very great deal of loneliness. This may, of course, be partly due to jealousy of those who are academically superior on one particular point, although they are inferior on a great many other points, because there is a certain amount of truth in the gibe that a specialist is a person who has failed in every subject except one, having had neither time nor energy to succeed in the others.
On Some General Ovals of Cassinian Type
Authors: Green H. Gwynedd
Published: January 2007

The following discussion of the curves r1r2 = crn, where r1, r2, r are the distances of a point from the “foci” ±α, 0 and the origin, c a positive constant and n rational (integral or fractional, positive or negative), provides a good example of extensive knowledge of general curves obtained by elementary-considerations. The attention of the author was first drawn to these curves by the remark of a colleague, Mr. N Davy, M.Sc., that the curves r1r2=cr⅓ appeared in the design of the pole-pieces of certain electromagnets.
Correlation of the Teaching of Geography and Mathematics
Authors: Fairgrieve J
Published: January 2007

There is a healthy attempt to break down the barriers that have hitherto existed between the subjects that are taught in schools. At the same time it must, I think, be taken as an axiom that the several major subjects of the curriculum, especially in secondary schools, should be organised without reference to each other if each is to make use of its own distinctive discipline. A complicated articulation would be unworkable.
Non-Associate Powers and a Functional Equation
Authors: Etherington I. M. H.
Published: January 2007

Several writers have studied algebras in which multiplication is non-associative, that is, x yz≠xy z. It is necessary in a non-associative algebra to distinguish the possible interpretations of a power xn In a non-commutative non-associative algebra x2 is unique, x3 can mean xx2 or x2x; x4 can mean x xx2, x x2x, x2x2, xx2x or x2x x, x5 has 14 interpretations; x6 has 42; and so on. In a commutative non-associative algebra, the possible interpretations are fewer x3 is unique, x4 can mean xx3 or x2x2, x5 can mean x xx3, x x2x2 or x2x3, x6 has 6 interpretations, and so on. The problem considered here is how many meanings are there for xn (A) in a general non-commutative non-associative algebra ? (B) in a general commutative non-associative algebra ? The answer to (A) is I am not able to find any such simple formula for (B).
The Importance of the History of Mathematics in Relation to the Study of Mathematical Technique
Authors: Whitrow G. J.
Published: January 2007

Glaisher, I believe, once said that no subject loses more than mathematics by any attempt to dissociate it from its history. It is still customary, however, in schools and most universities to regard the history of mathematics as an “extra” outside the ordinary student’s syllabus—indeed it is usually treated as an exotic luxury. The point I would like to emphasise is that while the primary aim of the tyro should be the mastery of technique, yet in most cases this mastery is liable to become a mere mechanical acquirement unless it is accompanied by some appreciation of the cultural significance of the “Queen of the Sciences”.
Place of Logic in the Teaching of Geometry
Authors: Black M.
Published: January 2007

Of all the reasons urged in favour of the teaching of mathematics and the favoured position of mathematics in the secondary school curriculum, the most widespread is the belief that to learn mathematics is valuable as a training in the art of reasoning. In this paper I want to explore the implications of this belief and its bearing upon the teaching of geometry I shall try to avoid a pronouncement on whether the belief in the value of mathematics as a training in reasoning is justified, by adopting the mathematical method of exploring the consequences of an assumption.
Celestial Distances
Authors: Cox R. R. S.
Published: January 2007

When asked to give this lecture I was
given to understand that this Society would like a
change from the discussion of pedagogical methods
and school problems, and could do with something
that had nothing to do with teaching at all. In
other words, I was to talk about stars and not to
mention schools. Well, I promise to do that,
though I must confess that the fact that members
of the Society are teachers was at the back of my
mind when choosing the subject. The schoolboy and
schoolgirl and the man in the street know that
celestial distances are very large; in fact the
adjective “astronomical” has been sadly degraded
by being applied to such mundane affairs as the
figures of the national debt—and a schoolboy of
enquiring mind may ask how the astronomers measure
such distances. Such an enquiry is often tinged
with a certain scepticism—sometimes politely
veiled, sometimes not. When you tell a member of
the public that a certain nebula is many million
light years away and that a light year is 6
billion miles, he is apt to adopt the attitude of
mind indicated by the phrase, “You’re telling me”,
and to wonder what meaning such a monstrous
distance can have: and how does the astronomer
know, anyway?
Vital Mathematics
Authors: Williamson R. S.
Published: January 2007

Mathematics existed prior to the
mathematician. Before Man calculated, Nature
multiplied and divided; her products were based on
geometrical forms. Mathematics, in reality, is not
a creation of the mathematician. It is a vital
spirit, immanent in Nature and Man, and expressed
in the forms and processes of Nature and the
manifold works of Man. This spirit functions only
partly through the mathematician, who limits both
its sphere of operation and its method of working.
These limitations constitute a serious difficulty
for the pupil by cramping the expression of his
personality. His love of beauty is ignored, his
social sympathy is untouched, his sense of humour
lies dormant, his interest in Nature is not
aroused, his imagination has little outlet, his
artistic, musical and religious tendencies find no
contact. On the whole his emotions are starved,
his intellectual development is limited and his
spiritual life is uninfluenced. These are hardly
the conditions for the successful teaching of any
subject. Nor does such limited mathematics
contribute adequately to the general culture of
the pupil. Its limitations are also a serious
obstacle to the integration of the school
curriculum, one of the most pressing
problems.
Theorems on the Tetrahedron
Authors: Robinson R. T.
Published: January 2007

Through any point 0 in space a straight line POP1 can be drawn meeting the opposite edges AD, BC of a tetrahedron ABCD in the points P, P 1 respectively; another straight line QOQ1 can be drawn to meet DB, AC in Q, QI; and another straight line ROR1 to meet DC, AB in R, RI.
Formulae for Numerical Differentiation
Authors: Bickley W G.
Published: January 2007

In a recent paper (1) formulae were given for the numerical integration of a function in terms of its values at a set of arguments at equal intervals. In this companion paper, formulae for numerical differentiation, using the same data, are collected. Their utility in enabling derivatives of a function given numerically at such a set of arguments to be computed is obvious, the need arises in several approximate methods which are coming more and more into use (2), (3). The formulae avoid the labour of preliminary differencing, and are indeed more convenient than the finite difference formulae when the derivative is required at all the points of subdivision of a limited range.
The Course as Seen from the University
Authors: Newman M. H. A.
Published: January 2007

Although I am to speak “from the University point of view” I wish to consider not only how the small number of boys who will go to a University may best be prepared for their work there, but also the general problem of the teaching of Algebra in the upper forms of schools. This problem is connected with a larger question, whether the balance of subjects is right in present-day school teaching in mathematics, and perhaps you will permit me to start with a short digression on this point.
Infinity in Euclidean Geometry
Authors: Green H. Gwynedd
Published: January 2007

It has been the fashion in some branches of mathematics to use a rather high sounding and mystifying terminology for comparatively simple ideas and I fear that Geometry has fallen to the lure. It is, of course, essential that new conceptions should have new titles. There is, however, the very serious danger that, through familiarity and, perhaps, moments of carelessness more may be read into such titles than was originally intended. I propose to discuss Infinity in Euclidean Geometry, as I find continually, especially with young students, that the phrases “point at infinity” and “line at infinity” have led to concepts of concrete existence which, whatever may be the case in other philosophies and branches of science, are in this one quite unjustified and which would be exceedingly awkward if they were correct.