propounded, at the beginning of his treatise, under the name of opoi, the definitions with which modern readers are familiar j under the name of atr^/xara, the three principles of construction now called postulates, together with the three theoretic principles, specially geometrical, now printed as the tenth, eleventh, and twelfth axioms ; finally, under the name of KOWXI ewoiat, or common notions, the series of general assertions concerning equality and inequality, having an application to discrete as well as continuous quantity, now printed as the first nine axioms. Now, throughout the Elements, there are numerous indications that Euclid could not have been acquainted with the logical doctrines of Aristotle : a most important one has been signalised in the article ANALYSIS, and, in general, it may suffice to point out that Euclid, who is said to have flour ished at Alexandria from 323 (the year of Aristotle s death) to 283 B.C., lived too early to be affected by Aristotle s work all the more that he was, by philosophical profession, a Platonist. Yet, although Euclid s disposition of geome trical principles at the beginning of his Elements is itself one among the signs of his ignorance of Aristotle s logic, it would seem that he had in view a distinction between his postulates and common notions not unlike the Aristotelian distinction between cu-n^ia-a and diuyxaTa. All the postulates of Euclid (including the last three so-called axioms) may be brought under Aristotle s description of atr^ftara principles concerning which the learner has, to begin with, neither belief nor disbelief, Post. Anal., i. 10, G) ; being (as De Morgan interprets Euclid s meaning) such as the " reader must grant or seek another system, whatever be his opinion as to the propriety of the assumption." Still closer to the Aristotelian conception of axioms come Euclid s common notions, as principles " which there is no question every one will grant" (De Morgan). From this point of view, the composition of Euclid s two lists, as they originally stood, becomes intelligible : be this, however, as it may, there is evidence that his enumeration and division of principles were very early subjected to criticism by his followers with more or less reference to Aristotle s doctrine. Apollonius (250-220 B.C.) is mentioned by Proclus (Com. in EucL, i.) as having sought to give demonstrations of the common notions under the name of axioms. Further, according to Proclus, Geminus made the distinction between postulates and axioms which has become the familiar one, that they are indemonstrable principles of construction and demonstration respectively. Proclus himself (412-485 A.D.) practically comes to rest in this distinction, and accordingly extrudes from the list of postulates all but the three received in modern times. The list of axioms he reduces to five, striking out as derivative the two that assert in equality (4th and 5th), also the two that assert equality between the doubles and halves of the same respectively (Gth and 7th). Euclid s postulate regarding the equality of right angles and the other assumed in the doctrine of parallel lines, now printed as the llth and 12th axioms, he holds to be demonstrable : the 10th axiom (regarded as an axiom, not a postulate, by some ancient authorities, and so cited by Proclus himself) Two straight lines cannot enclose a space he refuses to print with the others, as being a special principle of geometry. Thus he restricts the name axiom to such principles of demonstration as are common to the science of quantity generally. These, he then declares, are principles immediate and self-manifest untaught anticipations whose truth is darkened rather than cleared by attempts to demonstrate them. The question as to the axiomatic principles, whether of knowledge in general or of special science, remained where it had thus been left by the ancients till modern times, when new advances began to be made in positive scientific inquiry and a new philosophy took the place of the peri patetic system, as it had been continued through the Middle Ages. It was characteristic alike of the philosophic and of the various scientific movements begun by Des cartes to be guided by a consideration of mathematical method that method which had led in ancient times to special conclusions of exceptional certainty, and which showed itself, as soon as it was seriously taken up again, more fruitful than ever in new results. To establish philo sophical and all special truth after the model of mathe matics became the direct object of the new school of thought and inquiry, and the first step thither consisted in positing principles of immediate certainty whence deduction might proceed. Descartes accordingly devised his criterion of perfect clearness and distinctness of thought for the determination of ultimate objective truth, and his followers, if not himself, adopted the ancient word axiom for the principles which, with the help of the criterion, they proceeded freely to excogitate. About the same time the authority of all general principles began to be considered more explicitly in the light of their origin. Not that ever such consideration had been wholly over looked, for, on the contrary, Aristotle, in pronouncing the principles of demonstration to be themselves indemon strable, had suggested, however obscurely, a theory of their development, and his followers, having obscure sayings to interpret, had been left free to take different sides on the question ; but, as undoubtedly the philosophic investiga tion of knowledge has in the modern period become more and more an inquiry into its genesis, it was inevitable that principles claiming to be axiomatic should have their pretensions scanned from this point of view with closer vision than ever before. Locke it was who, when the Cartesian movement was well advanced, more especially gave this direction to modern philosophic thought, turn ing attention in particular upon the character of axioms ; nor was his original impulse weakened rather it was greatly strengthened by his followers substitution of positive psychological research for his method of general criticism. The expressly critical inquiry undertaken by Kant, at however different a level, had a like bearing on the question as to the nature of axiomatic principles ; and thus it has come to pass that the chief philosophic interest now attached to them turns upon the point whether or not they have their origin in experience. It is maintained, on the one hand, that axioms, like other general propositions, result from an elaboration of particular experiences, and that, if they possess an excep tional certainty, the ground of this is to be sought in the character of the experiences, as that they are exceptionally simple, frequent, and uniform. On the other hand, it is held that the special certainty, amounting, as it does, to positive necessity, is what no experience, under any circum stances, can explain, but is conditioned by the nature of human reason. More it is hardly possible to assert gene rally concerning the position of the rival schools of thought, for on each side the representative thinkers differ greatly in the details of their explanation, and there is, moreover, on both sides much difference of opinion as to the scope of the question. Thus Kant would limit the application of the name axiom to principles of mathematical science, denying that in philosophy (whether metaphysical or natural), which works with discursive concepts, not with intuitions, there can be any principles immediately certain ; and, as a matter of fact, it is to mathematical principles only that the name is universally accorded in the language of special science not generally, in spite of Newton s lead, to the laws of motion, and hardly ever to scientific principles of more special range like the atomic theory. Other thinkers, however, notably Leibnitz, lay stress on the ultimate prin
ciples of all thinking as the only true axioms, and would
