1 of the figure. Thus to-day the nine main problems and the numerous special cases, which are the subject-matter of 83 theorems in the two books de sectione determinata (of Pappus), constitute only one problem which can be solved by one single equation." "If we compare a mathematical problem with a huge rock, into the interior of which we desire to penetrate, then the work of the Greek mathematicians appears to us like that of a vigorous stonecutter who, with chisel and hammer, begins with indefatigable perseverance, from without, to crumble the rock slowly into fragments; the modern mathematician appears like an excellent miner, who first bores through the rock some few passages, from which he then bursts it into pieces with one powerful blast, and brings to light the treasures within." 2 ROME Although the Romans excelled in the science of government and war, in philosophy, poetry, and art they were mere imitators. In mathematics they did not even rise to the desire for imitation. If we except the period of decadence, during which the reading of Euclid began, we can say that the classical Greek writers on geometry were wholly unknown in Rome. A science of geometry with definitions, postulates, axioms, rigorous proofs, did not exist there. A practical geometry, like the old Egyptian, with empirical rules applicable in surveying, stood in place of the Greek science. Practical treatises prepared by Roman surveyors, called agrimensores or gromatici, have come down to us. 1 CHASLES, p. 39. "As regards the geometrical part of 2 HERMANN HANKEL, Die Entwickelung der Mathematik in den letzten Jahrhunderten, Tübingen, 1884, p. 9. these pandects, which treat exhaustively also of the juristic and purely technical side of the art, it is difficult to say whether the crudeness of presentation, or the paucity and faultiness of the contents more strongly repels the reader. The presentation is beneath the notice of criticism, the terminology vacillating; of definitions and axioms or proofs of the prescribed rules there is no mention. The rules are not formulated; the reader is left to abstract them from numerical examples obscurely and inaccurately described. The total impression is as though the Roman gromatici were thousands of years older than Greek geometry, and as though the deluge were lying between the two." Some of their rules were probably inherited from the Etruscans, but others are identical with those of Heron. Among the latter is that for finding the area of a triangle from its sides [the "Heronic formula"] and the approximate formula, & a2, for the area of equilateral triangles (a being one of the sides). But the latter area was also calculated by the formulas (a2 + a) and a2, the first of which was unknown to Heron. Probably a2 was derived from an Egyptian formula. The more elegant and refined methods of Heron were unknown to the Romans. The gromatici considered it sometimes sufficiently accurate to determine the areas of cities irregularly laid out, simply by measuring their circumferences. Egyptian geometry, or as much of it as the Romans thought they could use, was imported at the time of Julius Cæsar, who ordered a survey of the whole empire to secure an equitable mode of taxation. From early times it was the Roman practice to divide land into rectangular and rectilinear parts. Walls and streets were parallel, enclosing squares of prescribed dimensions. This practice 1 HANKEL, pp. 295, 296. For a detailed account of the agrimensores, consult CANTOR, I., pp. 485-551. 2 HANKEL, p. 297. simplified matters immensely and greatly reduced the necessary amount of geometrical knowledge. Approximate formulæ answered all ordinary demands of precision. Cæsar reformed the calendar, and for this undertaking drew from Egyptian learning. The Alexandrian astronomer Sosigenes was enlisted for this task. Among Roman names identified with geometry or surveying, are the following: Marcus Terentius Varro (about 116-27 B.C.), Sextus Julius Frontinus (in 70 A.D. prætor in Rome), Martianus Mineus Felix Capella (born at Carthage in the early part of the fifth century), Magnus Aurelius Cassiodorius (born about 475 A.D.). Vastly superior to any of these were the Greek geometers belonging to the period of decadence of Greek learning. It is a remarkable fact that the period of political humiliation, marked by the fall of the Western Roman Empire and the ascendancy of the Ostrogoths, is the period during which the study of Greek science began in Italy. The compilations made at this time are deficient, yet interesting from the fact that, down to the twelfth century, they were the only sources of mathematical knowledge in the Occident. Foremost among these writers is Anicius Manlius Severinus Boethius (480?-524). At first a favourite of King Theodoric, he was later charged with treason, imprisoned, and finally decapitated. While in prison he wrote On the Consolations of Philosophy. Boethius wrote an Institutio Arithmetica (essentially a translation of the arithmetic of Nicomachus) and a Geometry. The first book of his Geometry is an extract from the first three books of Euclid's Elements, with the proofs omitted. It appears that Boethius and a number of other writers after him were somehow led to the belief that the theorems alone belonged to Euclid, while the proofs were interpolated by Theon; hence the strange omission of all demonstration. The second book in the Geometry of Boethius consists of an abstract of the practical geometry of Frontinus, the most accomplished of the gromatici. Notice that, imitating Nicomachus, Boethius divides the mathematical sciences into four sections, Arithmetic, Music, Geometry, Astronomy. He first designated them by the word quadruvium (four path-ways). This term was used extensively during the Middle Ages. Cassiodorius used a similar figure, the four gates of science. Isidorus of Carthage (born 570), in his Origines, groups all sciences as seven, the four embraced by the quadruvium and three (Grammar, Rhetoric, Logic) which constitute the trivium (three path-ways). MIDDLE AGES ARITHMETIC AND ALGEBRA Hindus SOON after the decadence of Greek mathematical research, another Aryan race, the Hindus, began to display brilliant mathematical power. Not in the field of geometry, but of arithmetic and algebra, they achieved glory. In geometry they were even weaker than were the Greeks in algebra. The subject of indeterminate analysis (not within the scope of this history) was conspicuously advanced by them, but on this point they exerted no influence on European investigators, for the reason that their researches did not become known in the Occident until the nineteenth century. In India had no professed mathematicians; the writers we are about to discuss considered themselves astronomers. To them, mathematics was merely a handmaiden to astronomy. view of this it is curious to observe that the auxiliary science is after all the only one in which they won real distinction, while in their pet pursuit of astronomy they displayed an inaptitude to observe, to collect facts, and to make inductive investigations. It is an unpleasant feature about the Hindu mathematical treatises handed down to us that rules and results are expressed in verse and clothed in obscure mystic language. To him who |