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tetrakaidekahedron of wbich the twenty-four corners are the corners of the tetrahedrons thus placed may conveniently be called the orthic tetrakaidekahedron. It has six equal square faces and eight equal equiangular and equilateral hexagonal faces. It was described in $12 of my paper on “The Division of Space with Minimum Partitional Area, "* under the name of “plane-faced isotropic tetrakaidekahedron"; but I now prefer to call it orthic, because, for each of its seven pairs of parallel faces, lines forming corresponding points in the two faces are perpendicular to the faces, and the planes of its three pairs of square faces are perpendicular to one another. Fig. 8 represents an orthogonal projection on a plane parallel to one of the four pairs of hexagonal faces. The heavy lines are edges of the tetrakaidekahedron. The light lines are edges of the tetrahedrons of $ 13, or parts of those edges not coincident in projection with the edges of the tetrakaidekahedron. The figures 1, 1, 1; 2, 2, 2;. . .; 6, 6, 6 show corners belonging respectively to the six tetrahedrons,
Phil. Mag.,' 1887, 2nd half-year, and 'Acta Mathematica,' vol. 11, pp. 121-134.
two of the four corners of each being projected on one point in the diagram. Fig. 9 shows, on the same scale of magnitude with corresponding distinction between heavy and light lines, the orthogonal projection on a plane parallel to a pair of square faces.
§ 16. If the rule of $ 15 with reference to the division of each arm of a skeleton tetrahedron into four equal parts by points in which it is cut by other lines of skeletons is fulfilled with all details of $$ 14 and 15 applied to any oblique parallelepiped, we find a tetrakaidekahedron which we may call orthoid, because it is an orthic tetrakaidekahedron, altered by homogeneous strain. Professor Crum Brown has kindly made for me the beautiful model of an orthoidal tetrakaidekahedron thus defined which is placed before the Royal Society as an illustration of the present communication.
Fig. 10 is a stereoscopic picture of an orthic tetrakaidekahedron, made by soldering together thirty-six pieces of wire, each 4 in. long, with three ends of wire at each of twenty-four corners.
§ 17. I cannot in the present communication enter upon the most general possible plane-faced partitional tetrakaidekahedron or show its relation to orthic and orthoidal tetrakaidekahedrons. I
merely say that the analogy in the homogeneous division of a plane is this :-an equilateral and equiangular hexagon (orthic); any other hexagon of three pairs of equal and parallel sides whose paracentric diagonals trisect one another (orthoidal). The angles of an orthoidal hexagon, other than equilateral, are not 120°. The angles of the lefthand hexagon fig. 1 ($ 7) are 120°, and its paracentric diagonals do not trisect one another, as the diagram clearly shows.
II. “ An Estimate of the Degree of Legitimate Natality, as
shown in the Table of Natality compiled by the Author from Observations made at Budapest." By JOSEPH Körösi, Member of the Hungarian Academy of Sciences, Director of Municipal Statistics. Communicated by Sir JAMES PAGET, F.R.S. Received December 28, 1893.
(Abstract.) Both branches of the science of demography-natality statistics as well as mortality statistics-originated on British soil. It was in 1665 that the Royal Society published the first work on these matters (Graunt's “Observations "), whilst in 1693 Halley, by establishing the first life table, laid the foundation of the scientific treatment of mortality statistics. These tables of mortality showed for the first time how to measure the probability of death for each year of human age. The other branch of vital statistics is still in want of a corresponding table of natality, showing the probability of birth for each of the age-combinations of the parents. The table of natality is not of so great scientific importance as the life table, as the probability of death depends on natural laws, whilst the fertility, at least partially, is influenced by voluntary causes also. But as the problems of over-population or de-population are an effect of both forces, it is worth while to study the law of these facts also.
To reach this aim, I have tabulated the age of the 71,800 married couples given in the Census of 1891, conforming to the single yearcombinations. The virtual number of these combinations--as 45
productive years of the male have to be combined with each of the 40 productive years of the female—is about 2000. Knowing thus the number of all age-combinations, I observed for four years (two before and two after the Census) the 46,931 births amongst couples of those ages; I got thus, dividing the figures obtained by four, the yearly probability of birth for each age-combination.
As the legitimate natality is to be regarded as a resultant between two distinct forces, the instinct of nature which urges towards multiplication and the forethought which causes moral restraint, it was also desirable to get an insight into the march of the physiological fertility alone. For this purpose I had to look out for couples in whom the moral restraint is weakest or entirely absent. These conditions are fulfilled in newly-married couples, where the physiological factor is nearly exclusively active. I also thus found the degree of physiological natality for different age-combinations. The two curves are very different. The legitimate natality declines after the first child and its fall shows a regular slope. The physiological arrives later at its maximum, remains for some length of time on this culminating height, and decreases only at a more advanced age.
We get thus two degrees of fertility for each age. The difference between the degree of physiological and that of the actual fertility shows, the few cases of procreative exhaustion excepted, the influence of the moral factor. In the somewhat advanced ages this moral restraint exercises an influence exceeding all expectation. With the mothers of 30 to 35 it reduces the fertility to 78 per cent. (instead of 100 per cent.), with those of 43 to 2 per cent., i.e., 98/100 of the physiological faculty is suppressed. With men the influence is also very great, though weaker than with women.
Out of a large number of data here follow some figures to characterise the results :
For the mother.
For the father.
The tables containing the fertility of each sex separately I call monogenous, the others bigenous. Here follow some samples of the bigenous tables :
For husbands of 39 years of age the probability of becoming fathers
27 p.c. 17 ,
with a wife of 35 years.
21, I attach a volume of tables containing all the probabilities found, and an atlas of diagrams to make this heavy mass of figures more "accessible.