From the January 23, 1937, issue


Slums, and the manifold evils that they breed, are no new thing under America’s sun. Slumlike conditions obtained among the Pueblo Indians of the Southwest when white men first saw them—had existed, indeed for nearly half a thousand years before the first exploring parties of Spaniards penetrated into what is now the state of Arizona.

And these same ancient-American slum ways of living have been in large measure responsible for the decline of the Pueblo population; once 10 times more numerous than it is today in the uplands of northern Arizona. So at least declares Dr. Harold S. Colton, director of the Museum of Northern Arizona, at Flagstaff.

Slum life killed off the Pueblos in exactly the same way it kills off the wretched poor who house miserably in the East Sides of the white man’s cities, Dr. Colton charges. Crowded indecently together, ignorant of the elements of sanitation, lacking the means to practice it even if they had the knowledge, the people swallow disease-polluted water. And so they die—especially the children.


The famous theory of cosmogony of the expanding universe, which postulated some primeval explosion that sent the stars and galaxies rushing apart, has received a serious blow. The attack occurred in scientific discussions honoring the memory of the Notre Dame chemist and botanist, Father J.A. Nieuwland, whose basic discoveries led to the development of synthetic rubberlike compounds.

Prof. Arthur Haas, famous Viennese theoretical physicist now on the staff at Notre Dame University, presented mathematical arguments and calculations showing that the famous observed red-shift of light from the distant nebula can hardly be due to an expansion, or rushing away, of these cosmic bodies from some central point. The interpretation of the red-shift as due to a velocity of motion has been the backbone of the expanding-universe theory so often associated with the name of Abbé G. Lemaitre, noted Belgian scientist-priest.

Prof. Haas calculated the amount of energy which matter can create in a unit volume of the universe and finds it far too little to overcome the gravitational attraction that must be overcome if the different parts of the known universe are rushing away from one another in a superexpansion.

This new attack on the expanding universe comes close on the heels of the recent statement of Mt. Wilson Observatory’s famed astronomer, Dr. Edwin Hubble, who admitted in a Carnegie Institution of Washington lecture that one could now either consider the universe expanding or not; with perhaps a shade of evidence on the side of the not.

Prof. Haas backed up his arguments by also making calculations showing the amount of energy per gram of matter that would be required to double the mutual distances between nebulae in a system of mutual density. His conclusions from this calculation were:

“A nebular system which exhibits the average mass density observed in the extra-galactic region and which has already experienced a doubling of its linear dimensions, cannot possess a radius of more than about 6,000 million light years. This result seems remarkable because, if in the nebular system there is any expansion at all, we must assume that at least a doubling of the linear dimensions has already taken place. This conclusion cannot be evaded since a doubling requires only 1,300 million years, whereas the age of some terrestrial minerals was found to be 2,000 million years.”

If astronomers abandon the idea that the reddening of the light from the distant nebula is due to their velocity of expansion, then some other concept must be introduced to account for the observed red-shift.

Somehow or other, the light from these faraway nebula has to lose energy en route so that its color is slightly redder than it was when it started. One idea would be that intergalactic dust in the path would absorb some of the light-ray energy.

But Prof. Haas demonstrated by calculation that the loss of radiant energy is of the same order of magnitude as the energy production of matter. And he concluded by showing that every photon of radiant energy—whether it has high energy and short wavelength like an X ray or low energy and a long wavelength like a radio wave—loses the same energy in traveling one single wavelength. This concept would account for the observed red-shift in light from distant cosmic sources.


Hailed as one of the greatest recent advances in the science of numbers, Prof. Leonard Eugene Dickson of the University of Chicago has produced the first rigorous proof of an extension of one of the problems that has wrinkled the brows of mathematicians since the Middle Ages.

Ranking with the famous and impossible trisection of the angle as a brain puzzler, the task Prof. Dickson set himself and solved is what is called “additive number theory” or the “Waring problem.”

In its simplest form, the one that was discussed during the Middle Ages, the problem concerns the fact that every whole number is either an exact square or the sum of two, three, or four squares. By a coincidence, the famous mathematician Fermat in 1636—the year of the founding of Harvard, which is now being celebrated—first discovered the general theorem.

Many of the best brains in the world have set themselves the task of working out the rules, formulae, and proofs, and as early as 1772, a mathematician named Euler—son of a more celebrated mathematician—worked out the formulae for any power.

Amateur mathematicians may wish to ponder over it. Here it is. To express any number as the sum of two other numbers raised to any selected power, for convenience designated mathematically by the small number n, the maximum number needed of numbers so raised to the selected power is found by raising two to the selected power, subtracting two, and then adding the fraction three over two raised to the selected power, discarding the decimal fraction.

For squares the answer is four, for cubes it is nine, and for fourth powers it is 19, for fifth powers it is 37, and so on.

Mathematicians know and have confidence in this rule but it had never been rigorously proved for any but squares and cubes.

Prof. Dickson’s achievement is to prove it rigorously for all powers from the seventh power to infinity powers.

How did he do it? He did not even try to tell in the one lecture he gave. He explained that it would take 120 lectures to mathematically trained listeners to give full proof.

There are still three powers in additive number theory that have not yet been conquered, the fourth, fifth, and sixth powers. Prof. Dickson believes that, given time, he will work out the proof of these also.

Prof. Dickson glories somewhat in the impracticality of this particular branch of mathematics. It has been useful in the mathematics of the new quantum theory of physics, wave mechanics, and so on. But it hardly is useful as yet to practical chemists, physicists, and engineers who apply science to everyday life. That does not mean that it will not be useful in the future.

Going back to the formula for a minute, Prof. Dickson on the back of an envelope worked out the maximum number of terms in a series of seventh powers that will add up to any number. It is 143. Got a pencil and paper? You can work it out for yourself.

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