as noted by Descartes in 1638. Frenicle observed in 1657 that p must be of the form 4n+1. Sylvester proved in 1888 that no odd perfect number has fewer than six distinct prime factors, nor fewer than eight unless it be divisible by 3. A multiply perfect number, like 120 or 30240, is one the sum 3.120 or 4.30240 of whose divisors equals an exact multiple of the number. Such numbers, mostly very large, were found by Mersenne, Fermat, Frenicle, and Descartes in the years 1631-1647. - To obtain an interesting generalization of both perfect and amicable numbers, let n denote the sum of the divisors <n of n, let n, denote the sum of the divisors <n of n, etc. In case n is a perfect number, n=n1=n=➖➖, and conversely. If n. n, then n and n, are amicable numbers. If n ̧n, the chain of numbers n, n1, në, is said to be of period k. In 1913 the writer proved that there exists no chain of period 3, 4, 5 or 6 with n <6233, and verified for an extensive set of values of n that any non-periodic chain contains a prime, so that the next term is unity. By a more extended search, Poulet discovered in 1918 the following chain of period 5: n=12496=24:11:71, n1=2;19:47, n=2*:967, and noted that 14316 leads a chain of 28 terms. The writer defined an amicable triple to be three numbers such that the sum of the aliquot divisors of each equals the sum of the remaining two numbers, and obtained the following two examples of amicable triples formed of distinct numbers: 3.896, 11.296, 359b (b=214.5.19.31:151); 293-337c, 5.16561c, 99371c (c=25.3.13). Lionnet defined a perfect number of the second kind to be a number which equals the product of its aliquot divisors. But the only such numbers are p3 and pq, where p and q are distinct primes, and the question is too simple to be amusing. In 1657 Fermat challenged the English arithmeticians to find a cube (other than 1 and 73) which when increased by the sum of its aliquot divisors becomes a square. The least answer appears to be the cube of 2.3.5.13.41.47, the sum of whose divisors is the square of 27-32.52.7x 13.17.29. He also demanded a square which when increased by the sum of its aliquot divisors becomes a cube; the simplest know answer is the square of 43098=2.3.11.653, the sum of whose divisors is the cube of 1729 7.13.19. John Wallis found more complicated solutions of each problem by a very long computation. For complete references to the literature on the present subjects see the writer's History of the Theory of Numbers, Carnegie Institution of Washington, Vol. I, 1919, pp. 1-58, 393-407. CLIMATE AND HEALTH, WITH SPECIAL REFERENCE TO THE UNITED STATES1 By Professor ROBERT DeC. WARD HARVARD UNIVERSITY PRESENT VIEWS REGARDING THE RELATIONS OF CLIMATE AND HEALTH THE HE modern view as to the part played by climate in relation to health is very different from that held in earlier days. The cause of most diseases was formerly sought directly in atmospheric conditions. These conditions, to state the present view very broadly, are now believed to be important in two ways. They may affect, more or less directly, the life, development and virulence of the micro-organisms which are the specific cause of disease, and they may strengthen or weaken the individual's power of resistance against the attacks of these organisms. The older views concerning the predominant and direct influence of climate have largely been replaced by the conviction that good hygiene is more important than climate alone, and that, in the matter of the influence of a change of climate as a preventive or restorative, a change of residence, habits, occupations, food, is usually of as much importance as, if not of more importance than, the actual change in atmospheric conditions. If pure air, good food, freedom from worry, time for rest, proper exercise, outdoor life and a congenial occupation are provided, many bodily and mental ailments yield to the treatment. Climate is by no means to be discarded as of no account. It affects our physical and mental condition and our bodily comfort. It may be dull, rainy and cheerless, or bright, sunny and exhilarating. It may tend to keep us indoors, or it may naturally tempt us to go out. Thus some climates are naturally avoided; others are sought out. The choice of a suitable climate must depend upon the disease to be dealt with and upon the individual concerned. WHAT IS A GOOD CLIMATE? What, then, is a "good climate?" This question can here be answered only in the most general way. The answer, in individual cases, must obviously depend upon a person's physical and mental condition; upon his own personal preferences, and upon the factors other than climate which should be taken into account in each special Part of a Presidential Address before the American Meteorological Society, Chicago, Ill., December 29, 1920. The writer is indebted to Dr. Guy Hinsdale, of Hot Springs, Va., for valued criticisms. case. A "good climate" is naturally one which is favorable to the development of a sturdy race of men and women, physically strong and mentally alert. There is a pretty general agreement, among physicians, physiologists and climatologists, that, excepting those who are distinctly ill, the best climate for most people and most of the time is one which has frequent moderate weather changes; fairly marked annual and diurnal variations in temperature; a reasonable amount of cold during at least part of the year; a refreshing variety in the amount of cloudiness, and sufficient rainfall to provide enough moisture for the growth of grass and crops. Such a climate is an intermediate one. It is neither invariably hot nor permanently cold. It is neither monotonously arid and cloudless, nor always dull and rainy. It is between all extremes. The climates of much of the so-called "temperate zones" are of this general type. Their physiological effects are intermediate between those of the equatorial and those of the polar zones. They exercise the body's power of reaction and adaptation, keeping it physiologically active, and in good "working condition," without subjecting the different organs to too severe a strain.2 A climate which encourages people to spend the maximum possible amount of their time outdoors, in the open air, is, other things being equal, the best for the majority of men and women. Applying these general principles to the question of health, it may be stated that a health resort where a patient can find comfortable quarters, congenial company, plenty of diversion, and where favorable climatic conditions such as those above noted, with abundant sunshine, an absence of disagreeable winds and dust and of sudden marked weather changes, encourage outdoor life, is to be recommended. The climate does not necessarily and inevitably cure, but it is very often an important help in the treatment of disease. To quote Sir H. Weber, "for any given class of cases, that climate is a good one in which the qualities that would be disadvantageous are to a certain degree absent during the whole year, or at least part of the year, while the other qualities are present by the proper use of which the bodily strength is raised and the restoration of the affected organs and functions is facilitated." IS THERE ANY "PERFECT" CLIMATE? In the foregoing quotation from one of the leading medical climatologists, emphasis is laid on one point concerning which there is a general and persistent misconception. "That climate is a good one," Sir H. Weber wrote, "in which the qualities that would be disadvantageous are to a certain degree absent during the whole year, or at least part of the year." In other words, a good climate has the fewest 2F. P. Weber and G. Hinsdale, "A System of Physiologic Therapeutics," Philadelphia, 1901, Vol. III, p. 18. "outs," or is free from its "outs" during a portion of the year. It is often said that the climate of a certain place is "perfect" or "ideal." As a matter of fact, there is no such thing as a "perfect" climate, anywhere, or all the time. Every climate has some disagreeable features. Health resorts are never equally desirable at all seasons. It is probably safe to say that every climate has advantages of its own for some special purpose, but some climates have more, and some have fewer, disadvantages. A southern climate which has a mild and genial "winter," and therefore provides abundant opportunities for an outdoor life at that season, is likely to be too hot, or rainy, or dusty, in summer. A northern climate, which has the advantages of cool summers, is likely to be very cold and stormy and snowy in winter. A western seacoast in subtropical latitudes, with the attractions of equable temperatures, mild winters and cool summers, relatively small rainfall and few stormy days, may be too damp and too foggy for many invalids. Mountain resorts, often so useful in the treatment of lung diseases, may have the disadvantages of being too cool at night; too windy and dusty by day, or of having frequent severe local storms. "Perfect" climates, then, do not exist. In climatotherapy, which may be broadly defined as the use of climate for checking or preventing the development of disease, and for aiding the recovery of those who are ill or convalescent, the obvious course is to select a locality where the other necessary conditions, such as suitable accommodations, good food, expert medical attendance, and so on, are already provided, and where the climate has a maximum of the desired characteristics for the particular case concerned, and the minimum of undesirable features. The seeker after health whose physician orders a "change of climate" should go away expecting some conditions which are neither "perfect," nor perhaps even altogether agreeable. If a "perfect" climate is not anticipated, the inevitable lack of perfection will bring with it no disappointment. It should be remembered that any climate which is recommended by a competent physician doubtless has far more good qualities than bad ones. The utmost advantage can always be taken of the good; most of the temporary disagreeable ones can be escaped by staying indoors, and by maintaining a cheerful disposition which refuses to be overcome by an occurrence of so little importance as an occasional spell of bad weather. NO CLIMATE IS "THE ONLY CLIMATE" FOR ANY INDIVIDUAL Case of ILL HEALTH In connection with this misconception regarding the existence of "perfect" climates, there is a widespread and persistent popular impression that certain climates have such special and peculiar properties of their own that there is some one particular climate which is the only one fitted for a special case of illness. If an invalid hits upon this climate, he believes that he will, beyond a doubt, regain his health. In my own experience of a score of years I have had abundant and often very distressing opportunity to learn how widespread is this notion, and how nerve-racking, in many people's minds, the choice of this one climate becomes. I have received many letters, personal visits and even telephone calls, asking me to name that special place whose climate is the one needed in the individual case concerning which the inquiry is made. It is hardly necessary for me to say that under no circumstances do I attempt-nor should any climatologist attempt to prescribe any climate for any specific disease. That is the responsibility of the physician. The most pathetic case which has ever come to my own attention was that described in a letter which I received a few years ago. The letter was from a physician in New York State, who had a son seriously ill with a complication of disorders. Leading specialists in New York had been consulted, but the youth was steadily growing worse, and there seemed little hope of saving his life. Evidently in depair, and as a last resort, the father, who was unknown to me even by name, wrote to me, giving me full details as to his son's illness and the previous treatment of the disease, and asked me to name the place to which the son could be taken in order to give him the benefit of whatever climate, in my judgment, would be the suitable one for that particular case. That letter revealed the whole story of the father's love for his boy; the tragedy of his despair of saving the boy's life; the grasping at a "last straw" in his writing to me as one of whom he had somehow or other heard and who might possibly give the advice which would save the son's life. What could I write in reply to that father's pathetic and despairing appeal? All I could do was to write a letter as full of human sympathy as it was possible to make it; to urge consultation with some physician who had made a specialty of climatotherapy; to point out the importance of selecting some place where every possible comfort and care could be found, and then to mention the general advantages of the winter climates of a considerable number of Southern resorts where relief from the winter cold and from the sudden and severe weather changes of New York State might be found. I need not say that I named no place as possessing the one climate which would be of most help in the curative process. The end of this story I do not know. No second letter ever came to me from that father. This incident serves as a striking illustration of exactly what I am here trying to emphasize. Climate, as a recent writer has well put it, may "play an important part in the curative process, but the climate of certain localities does not possess any peculiar properties which act as a specific on certain diseases." Or, as another has put it, "the choice of a climate is not a nerve-racking decision the entire success of which depends upon hitting upon the one ideal climate in the whole |