ON THE HISTORY OF "PLAYFAIR'S PARALLEL-POSTULATE."

BY FLORIAN CAJORI,

University of California.

The cumbersome form of the parallel-postulate given in Euclid's Elements has been generally displaced in modern textbooks by an equivalent postulate, often called "Playfair's Postulate": "Two straight lines which intersect one another cannot be both parallel to the same straight line." A different phrasing is the following: "Through a given point not more than one parallel can be drawn to a given straight line." Either form is much simpler than Euclid's statement: "If a straight line meet two straight lines so as to make the two interior angles on the same side of it taken together less than two right angles, these straight lines, being continually produced, shall at length meet on that side on which are the angles which are less than two right angles." Playfair himself laid no claim to the postulate as he gave it; in fact, he stated in the "Notes" to his edition. of Euclid that it "has been assumed by others, particularly by Ludlam, in his very useful little tract entitled Rudiments of Mathematics," 1785. We presume that Playfair quoted Ludlam as the earliest claimant known to him.

In view of the popularity of this postulate, not only in elementary texts but also in advanced treatises on the foundations of geometry, it may be of interest to cite an author using it who antedates both Playfair and Ludlam. This author is Joseph Fenn, who published an edition of Euclid's Elements in 1769 at Dublin. This Euclid is part of the "First Volume of the Instructions given in the Drawing School established by the Dublin-Society, persuant to their Resolution of the Fourth of February, 1768. Under the Direction of JOSEPH FENN, heretofore Professor of Philosophy in the University of Nants, Dublin, MDCCLXIX." The usual sources of biographical and historical information fail to mention either him or his books. He argues in his edition of Euclid that the postulate given by Euclid "is not simple enough" and proceeds to remedy matters by interpolating between the 28th and 29th propositions of the first book a lemma that, if a straight line meeting two straight lines makes the alternate interior angles unequal, those two straight lines meet. In proving this lemma he makes use of a principle which in the body of the proof is designated as "C. N." (common notion): "A straight line which cuts one of two parallels will necessarily cut the other, provided this cutting

line is sufficiently produced." This is the same as the assumption of Ludlam and Playfair. This principle, says Fenn, "Euclid regards as self-evident"; Euclid makes use of it, Fenn says, in the 30th and 37th propositions of the first book, where a straight line cutting one of two or three parallel lines is made to cut all of them. As a matter of fact, Euclid rightfully regarded this property of parallel lines as constituting an immediate consequence of his own parallel-postulate.

The frontispiece of this volume represents the Greek philosopher Aristippus (regarded as having been unfriendly to mathematical pursuits) as shipwrecked on the coast of the Rhodians. When he observes geometrical figures drawn upon the rocks, he cries to his companions in distress: "Let us be of good cheer, for I see traces of man," and thereby admits that mathematics stands for intelligence.

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ON THEORIES CONCERNING SOILS AS MEDIA FOR PLANT GROWTH.

BY CHAS. B. LIPMAN,

University of California, Berkeley.

(Continued from November Issue.)

These theories, together with the so-called laws of Liebig which I have just cited, were the cause of protracted controversies, some of which were not settled for a period of fifteen or twenty years. Indeed, one of the controversies (the nitrogen question) was not fully disposed of until 1885. The disputes in question were the result of experimental work in the field carried on by Lawes and later by Lawes and Gilbert in England, and beginning about the same time as the famous experiments of Boussignault in France. By actual results obtained on the analysis of the ash of certain plants and by the response of such plants to fertilizer ingredients, Lawes and Gilbert demonstrated that the composition of the ash of plants was no necessary index to the needs of plants for the mineral elements. They also showed that the air could not be depended upon to furnish sufficient nitrogen to plants when the soil was unequal to the task, reserving as incomprehensible the case of leguminous plants which usually proved to be exceptions to this rule. I wish that time and your patience permitted me to descant on the more detailed phases of those remarkable experiments of Lawes and Gilbert, wonderfully conceived, accurately planned, and scrupulously and ably executed, which served to answer and effectively dispose of Liebig's mineral theory and largely of his nitrogen theory; to tell of Liebig's patent fertilizer and the story and reason for its failure; and to describe the numerous investigations which these theories stimulated. It is impossible, however, to go into all these matters in the brief space of an hour. I must, therefore, content myself with what I have told you of Liebig's work and with the further comment that despite a number of errors which crept into Liebig's theories, the latter were essentially sound, and while they constituted very largely amplifications of de Saussure's theories and facts, they remain as great achievements from the point of view of scientific propaganda. The position in the annals of agricultural chemistry and in plant physiology occupied by Liebig, however, represents only a small part of the credit which is due him in the general field and particularly in the organic part of chemistry.

Before passing on to the progress of soil theory after 1860,

by which date the controversies between Lawes and Gilbert and Liebig were largely settled, there must be noted an old theory in regard to the cause of soil infertility, which has been of the greatest potency, directly or indirectly, in some of the most re-. cent developments in soil theory. I refer to a modification of a very old idea of the selective powers of plant roots introduced into more or less modern literature by the distinguished French plant physiologist, Decandolle, in 1840. According to this theory, the roots of plants excrete certain substances which on accumulation become toxic to the same variety of plant, but not necessarily to other varieties of plants. Hence, the theory goes on to account for the benefits of crop rotation. It is interesting that this view should have been revived for the first time contemporaneously with the appearance of Liebig's report to the British Association, and particularly that it should have been regarded favorably by Liebig as the latter's more or less detailed consideration of it indicates. I shall return to a consideration of the Decandolle theory later in connection with a recrudescence of it in a period close to our own.

Great progress was being made in the sixties and seventies of the last century in all branches of science and much of it had its important relation to and effect upon soil theory. The brilliant work of Pasteur was going on and was opening the way for the development of modern microbiology, which dates from the discovery, by Robert Koch in 1882, of the plate method of isolating bacteria. Pfeffer and De Vries independently made their celebrated contributions to science in 1877 on the subject of osmotic pressure with particular reference to absorption by plants, thus rendering clearer the mechanism of salt action in the medium of plant growth. Many other pieces of outstanding work of a lesser magnitude were being accomplished. But none contributed anything to modify materially the theory of plant nutrition which may be regarded as a composite of the views of de Saussure, Liebig, and Lawes and Gilbert, on which all of those investigators would have agreed. Nor did any of them shed any light on the puzzling question of the legumes in connection with the nitrogen nutrition of plants. A radical departure from established experimental procedure was again needed as in several periods of the history of soil theory before. It was soon forthcoming, however, with the development of general, and closely on the heels therefore, of soil bacteriology. As far back as 1862, Pasteur suggested that nitrification was probably a bacte

rial process. Up to then, and even for some time after that, it was regarded as a chemical reaction taking place without the inter vention of living organisms. In 1877, however, the year in which Pfeffer and De Vries made their important contributions to the subject of the osmotic character of the soil, Schloesing and Muntz, two French investigators, definitely demonstrated the process of nitrification to be of biological origin. They found that when sewage containing ammonia was poured very slowly through tubes of sand, the ammonia would come through unchanged for a period of twenty days, after which it would disappear partly or wholly and its place would be taken by nitrates. When chloroform was poured into the tubes, nitrates would soon cease to appear in the filtrate and again ammonia would be found. When the chloroform was allowed to evaporate and the sand reinfected with some fresh soil, nitrates would again be found in the filtrate in a few days. This demonstrated beyond a peradventure that nitrification was a process due to living agencies. It took thirteen years more, however, and the patient attention and sedulous industry of a number of leading chemists and physiologists, culminating in the isolation of the nitrifying bacteria by Winogradsky in 1890, to make clear the process by which nitrogen in soil organic matter is rendered into nitrates.

Even this splendid work on nitrification, including Winogradsky's classic contribution, however, would still have left unexplained the abnormal behavior of legumes in the experiments of Lawes and Gilbert. But the necessary explanation was made through the striking and epoch making discoveries of Hellriegel, and of Hellriegel and Wilfarth, made public in 1885, and soon confirmed by Lawes and Gilbert, and by many others. These investigators found that when planted in a sterile medium, legumes do not behave abnormally, but respond to soluble nitrogen like non-legumes, growth being more or less regularly increased with every increment of nitrate nitrogen. When, however, the medium was not sterile, they would at times behave like non-legumes and at times not. They found, further, that nodules containing large numbers of bacteria were always found on the roots of the plants behaving abnormally. They connected these various facts and decided that the bacteria in the nodules on the roots of legumes, by some mechanism, obtained access to the large store of nitrogen in the air, which was not available to non-legumes, or to non-infected legumes. The nodules had been known since 1687, and the bacterial bodies in them