The materials for a history of Greek geometry after Apollonius are both scanty in quantity and most unsatisfactory in quality. We know the names of many geometers who lived during the next three centuries, but very few indeed of their works have come down to us, and we are compelled to rely for the most part on such scraps of information as the later scholiasts, Pappus, Proclus, Eutocius and the like, have incidentally preserved. But this information, again, generally affords little clue to the date of the geometer in question. Thus, though we have abundant evidence that mathematics remained a chief constituent of the Greek liberal curriculum, we cannot tell with any accuracy what subjects were most in vogue or what mathematicians were most generally regarded at any particular time. It is certain, however, that during the whole period between Apollonius and Ptolemy only two mathematicians of real genius, Hipparchus and Heron, appeared, that both of these lived about the same time (120 b.c.), and that neither was interested in mathematics per se, for Hipparchus was above all things an astronomer, Heron above all things a surveyor and engineer. The result might have been different if some new methods had been introduced. The force of nature could go no further in the same direction than the ingenious applications of exhaustion by Archimedes and the portentous sentences in which Apollonius enunciates a proposition in conies. A briefer symbolism, an analytical geometry, an infinitesimal calculus were wanted, but against these there stood the tremendous authority of the Platonic and Euclidean tradition, and no discoveries were made in physics or astronomy which rendered them imperatively necessary.

In the book of Problemata, attributed to Aristotle, the following question is asked (xv. 3): “Why do all men, both barbarians and Hellenes, count up to 10 and not to some other number?” It is suggested, among several answers of great absurdity, that the true reason may be that all men have ten fingers: “using these, then, as symbols of their proper number (viz. 10), they count everything else by this scale.” The writer then adds “Alone among men, a certain tribe of Thracians count up to 4, because, like children, they cannot remember a long sum neither have they any need for a great quantity of anything.”

It is natural to regret that an author who at so early a date was capable of writing this passage, was not induced to ask himself more questions and to collect more facts on the same and similar subjects. Had he done so, he might have anticipated, by some two thousand years, the modern method of research into prehistoric times and might have attempted, with every chance of success, a hundred problems which cannot now be satisfactorily treated. In the fourth century B. c. and for long after, half the Aryan peoples were still barbarous and there must still have survived, even among Greeks and Italians, countless relics of primitive manners, forming a sure tradition of the past. Nearly all these materials, so abundant in Aristotle's day, are irretrievably lost to us and the primeval history of Aryan culture depends now chiefly on the evidence supplied by comparative philology. It is so with the art of calculation.

It has been already pointed out that the conditions of life in Athens were unfavourable to the growth of any “natural” science. Her practical men were absorbed in politics, her philosophers in metaphysical speculation. Neither of these classes objected to deductive science, for deduction is the chief instrument of rhetoric and is also the most interesting part of logic: but the patient and unrewarded industry, which leads to inductive science, was not to the Athenian taste. The practical men thought it profane, the philosophers vulgar. The schools of inductive science remained therefore far away from the turmoil of Athens: the observatories of the astronomers were at Cyzicus on the Hellespont or at Cnidus on the south coast of Asia Minor: the school of medicine was maintained by one illustrious family in the island of Cos. If it be objected that Aristotle lived in Athens, the answer is that Aristotle was the son of a physician, was not born or bred in Athens, never became an Athenian citizen, disliked Athens and left it, and was not able to command in Athens an audience for anything but metaphysics. The Peripatetic school was as unscientific as the Platonic. There was not yet a “university, ” to which all the world might come and learn all the knowledge that was in existence. Alexandria was the first city to deserve that name. Athens might have won it, but when Athenian politics were no more and the field was free for other pursuits, Alexandria had forestalled her.

The preceding pages contain probably all the meagre facts from which it is still possible to discern how the Greeks came by their arithmetical nomenclature, both for whole numbers and for fractions. The subsequent progress of calculation, that is to say, the further use of the elementary processes, depends on many conditions which cannot well be satisfied without a neat and comprehensive visible symbolism. This boon the Greeks never possessed. Yet even without it a retentive memory and a clear logical faculty would suffice for the discovery of many important rules, such for instance as that, in a proportion, the product of the means is equal to the product of the extremes. It is probable, therefore, that much of the Greek arithmetical knowledge dates from a time far anterior to the works in which we find historical evidence of it. It is probable, again, that the Greeks derived from Egypt at an early date as many useful hints on arithmetic as they certainly did on geometry and other branches of learning. It becomes necessary, therefore, to introduce in this place some account of Egyptian arithmetic, both as showing at what date certain arithmetical rules were known to mankind and as providing a fund of knowledge from which the Greeks may have drawn very largely in prehistoric times. The facts to be now stated are in any case of great importance, since they furnish the only compact mass of evidence concerning the difficulties which beset ancient arithmetic and the way in which they were surmounted.

Quite recently a hieratic papyrus, included in the Rhind collection of the British Museum, has been deciphered and found to be a mathematical handbook, containing problems in arithmetic and geometry.

An elaborate history of Greek geometry before Euclid was written by Eudemus, the pupil of Aristotle, who lived about 330 b.c. The book itself is lost but is very frequently cited by later historians and scholiasts, and it may be suspected also that many notices, not directly ascribed to it, were taken from its pages. Proclus, the scholiast to Euclid, who knew the work of Eudemus well, gives a short sketch of the early history of geometry, which seems unquestionably to be founded on the older book. The whole passage, which proceeds from a competent critic, and which determines approximately many dates of which we should otherwise be quite ignorant, may be here inserted verbatim by way of prologue. It will be cited hereafter as “the Eudemian summary.” It runs as follows:

“Geometry is said by many to have been invented among the Egyptians, its origin being due to the measurement of plots of land. This was necessary there because of the rising of the Nile, which obliterated the boundaries appertaining to separate owners. Nor is it marvellous that the discovery of this and the other sciences should have arisen from such an occasion, since everything which moves in development will advance from the imperfect to the perfect. From mere sense-perception to calculation, and from this to reasoning, is a natural transition. Just as among the Phoenicians, through commerce and exchange, an accurate knowledge of numbers was originated, so also among the Egyptians geometry was invented for the reason above stated.

17. A distinction is drawn, and very naturally and properly drawn, by the later Greek mathematicians between ὰριθμητική and λογιστική, by the former of which they designated the ‘science of numbers,’ by the latter, the ‘art of calculation.’ An opposition between these terms occurs much earlier and is frequently used by Plato, but though λογιστική can hardly mean anything but ‘calculation,’ it is not quite clear whether ὰριθμητική then bore the sense which it had undoubtedly acquired by the time of Geminus (say b.c. 50). That it did so, however, is rendered pretty certain by many circumstances. It is probable, in the first place, that the Pythagoreans would have required some variety of terms to distinguish the exercises of schoolboys from their own researches into the genera and species of numbers. In Aristotle a distinction, analogous to that between the kinds of arithmetic, is drawn between γεωδaισίa, the practical art of land-surveying, and the philosophical γεωμετρίa. Euclid, who is said to have been a Platonist and who lived not long after Plato, collected a large volume of the theory of numbers, which he calls ὰριθμητική only and in which he uses exactly the same nomenclature and symbolism as we find in those passages where Plato draws a philosophical illustration from arithmetic. It may therefore be assumed that λογιστική and ὰριθμητική covered, respectively, the same subjectmatter in Plato's time, as afterwards and since he uses these terms casually, with no hint that they were novel, we may infer that the distinction between them dates from a very early time in the history of Greek science and philosophy.

The earliest history of Geometry cannot be treated in the same way as that of Arithmetic. There is not for the former, as there is for the latter, a nomenclature common to many nations and languages; and the analysis of a geometrical name in any one language leads only to the discovery of a rootsyllable which is common to many very different words and to which only the vaguest possible meaning may be assigned. Nor is any assistance, so far as I know, furnished by travellers among savage and primitive races. Arithmetical operations are matters of such daily necessity that every general arithmetical proposition, of which a man is capable, is pretty certain to be applied in his practice and to attract attention: but a man may well know a hundred geometrical propositions which he never once has occasion to use, and which therefore escape notice. I have sought, in vain, through many books which purport to describe the habits and psychology of the lower races, for some allusion to their geometrical knowledge or for an account of some operations which seem to imply geometrical notions. One would be glad, for instance, to learn whether savages anywhere distinguish a right angle from an acute. Have they any mode of ascertaining whether a line is exactly straight or exactly circular? Do they by name distinguish a square from any other rectilineal figure? Do they attach any mysterious properties to perpendicularity, angular symmetry, etc.?

The history of Greek mathematics is, for the most part, only the history of such mathematics as are learnt daily in all our public schools. And very singular it is that, though England is the only European country which still retains Euclid as its teacher of elementary geometry, and though Cambridge, at least, has, for more than a century, required from all candidates for any degree as much Greek and mathematics together as should make this book intelligible and interesting, yet no Englishman has been at the pains of writing, or even of translating, such a treatise. If it was not wanted, as it ought to have been, by our classical professors and our mathematicians, it would have served at any rate to quicken, with some human interest, the melancholy labours of our schoolboys.

The work, as usual, has been left to Germany and even to France, and it has been done there with more than usual excellence. It demanded a combination of learning, scholarship and common sense which we used, absurdly enough, to regard as peculiarly English. If anyone still cherishes this patriotic delusion, I would advise him to look at the works of Nesselmann, Bretschneider, Hankel, Hultsch, Heiberg and Cantor, or, again, of Montucla, Delambre and Chasles, which are so frequently cited in the following pages. To match them we can show only an ill-arranged treatise of Dean Peacock, many brilliant but scattered articles of Prof. De Morgan, and three essays by Dr Allman.