tain over 50 different classes of problems, which are strung together without any attempt at classification. But still more multifarious than the problems are the solutions. General methods are unknown to Diophantus. Each problem has its own distinct method, which is often useless for the most closely related problems. "It is, therefore, difficult for a modern, after studying 100 Diophantine solutions, to solve the 101st."7

That which robs his work of much of its scientific value is the fact that he always feels satisfied with one solution, though his equation may admit of an indefinite number of values. Another great defect is the absence of general methods. Modern mathematicians, such as Euler, La Grange, Gauss, had to begin the study of indeterminate analysis anew and received no direct aid from Diophantus in the formulation of methods. In spite of these defects we cannot fail to admire the work for the wonderful ingenuity exhibited therein in the solution of particular equations.

It is still an open question and one of great difficulty whether Diophantus derived portions of his algebra from Hindoo sources or not.

THE ROMANS.

Nowhere is the contrast between the Greek and Roman mind shown forth more distinctly than in their attitude toward the mathematical science. The sway of the Greek was a flowering time for mathematics, but that of the Roman a period of sterility. In philosophy, poetry, and art the Roman was an imitator. But in mathematics he did not even rise to the desire for imitation. The mathematical fruits of Greek genius lay before him untasted. In him a science which had

no direct bearing on practical life could awake no interest. As a consequence, not only the higher geometry of Archimedes and Apollonius, but even the Elements of Euclid, were entirely neglected. What little mathematics the Romans possessed did not come from the Greeks, but from more ancient sources. Exactly where and how it originated is a matter of doubt. It seems most probable that the "Roman notation," as well as the practical geometry of the Romans, came from the old Etruscans, who, at the earliest period to which our knowledge of them extends, inhabited the district between the Arno and Tiber.

Livy tells us that the Etruscans were in the habit of representing the number of years elapsed, by driving yearly a nail into the sanctuary of Minerva, and that the Romans continued this practice. A less primitive mode of designating numbers, presumably of Etruscan origin, was a notation resembling the present "Roman notation." This system is noteworthy from the fact that a principle is involved in it which is not met with in any other; namely, the principle of subtraction. If a letter be placed before another of greater value, its value is not to be added to, but subtracted from, that of the greater. In the designation of large numbers a horizontal bar placed over a letter was made to increase its value one thousand fold. In fractions the Romans used the duodecimal system.

Of arithmetical calculations, the Romans employed three different kinds: Reckoning on the fingers, upon the abacus, and by tables prepared for the purpose.3 Finger-symbolism was known as early as the time of King Numa, for he had erected, says Pliny, a statue of the double-faced Janus, of which the fingers indicated 365 (355?), the number of days in a year. Many other passages from Roman authors point out the use of the fingers as aids to calculation. In fact, a fingersymbolism of practically the same form was in use not only in

Rome, but also in Greece and throughout the East, certainly as early as the beginning of the Christian era, and continued to be used in Europe during the Middle Ages. We possess no knowledge as to where or when it was invented. The second mode of calculation, by the abacus, was a subject of elementary instruction in Rome. Passages in Roman writers indicate that the kind of abacus most commonly used was covered with dust and then divided into columns by drawing straight lines. Each column was supplied with pebbles (calculi, whence 'calculare' and 'calculate') which served for calculation. Additions and subtractions could be performed on the abacus quite easily, but in multiplication the abacus could be used only for adding the particular products, and in division for performing the subtractions occurring in the process. Doubtless at this point recourse was made to mental operations and to the multiplication table. Possibly finger-multiplication may also have been used. But the multiplication of large numbers must, by either method, have been beyond the power of the ordinary arithmetician. To obviate this difficulty, the arithmetical tables mentioned above were used, from which the desired products could be copied at once. Tables of this kind were prepared by Victorius of Aquitania. His tables contain a peculiar notation for fractions, which continued in use throughout the Middle Ages. Victorius is best known for his canon paschalis, a rule for finding the correct date for Easter, which he published in 457 A.D.

Payments of interest and problems in interest were very old among the Romans. The Roman laws of inheritance gave rise to numerous arithmetical examples. Especially unique is the following: A dying man wills that, if his wife, being with child, gives birth to a son, the son shall receive and she of his estates; but if a daughter is born, she shall receive and his wife. It happens that twins are born, a boy and a girl.

How shall the estates be divided so as to satisfy the will? The celebrated Roman jurist, Salvianus Julianus, decided that the estates shall be divided into seven equal parts, of which the son receives four, the wife two, the daughter one.

We next consider Roman geometry. He who expects to find in Rome a science of geometry, with definitions, axioms, theorems, and proofs arranged in logical order, will be disappointed. The only geometry known was a practical geometry, which, like the old Egyptian, consisted only of empirical rules. This practical geometry was employed in surveying. Treatises thereon have come down to us, compiled by the Roman surveyors, called agrimensores or gromatici. One would naturally expect rules to be clearly formulated. But no; they are left to be abstracted by the reader from a mass of numerical examples. "The total impression is as though the Roman gromatic were thousands of years older than Greek geometry, and as though a 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 and the approximate formula, 18 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 the expression a2 was derived from the Egyptian formula a+b.c+d for the determina

3

2

2

tion of the surface of a quadrilateral. This Egyptian formula was used by the Romans for finding the area, not only of rectangles, but of any quadrilaterals whatever. Indeed, the gromatici considered it even sufficiently accurate to determine the areas of cities, laid out irregularly, simply by measuring their circumferences.7 Whatever Egyptian geometry the Romans possessed was transplanted across the Mediterranean at the

time of Julius Caesar, who ordered a survey of the whole empire to secure an equitable mode of taxation. Cæsar also reformed the calendar, and, for that purpose, drew from Egyptian learning. He secured the services of the Alexandrian astronomer, Sosigenes.

In the fifth century, the Western Roman Empire was fast falling to pieces. Three great branches-Spain, Gaul, and the province of Africa-broke off from the decaying trunk. In 476, the Western Empire passed away, and the Visigothic chief, Odoacer, became king. Soon after, Italy was conquered by the Ostrogoths under Theodoric. It is remarkable that this very period of political humiliation should be the one during which Greek science was studied in Italy most zealously. School-books began to be compiled from the elements of Greek authors. These compilations are very deficient, but are of absorbing interest, 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 Boethius (died 524). At first he was a great favourite of King Theodoric, but later, being charged by envious courtiers with treason, he was imprisoned, and at last decapitated. While in prison he wrote On the Consolations of Philosophy. As a mathematician, Boethius was a Brobdingnagian among Roman scholars, but a Liliputian by the side of Greek masters. He wrote an Institutis Arithmetica, which is essentially a translation of the arithmetic of Nicomachus, and a Geometry in several books. Some of the most beautiful results of Nicomachus are omitted in Boethius' arithmetic. The first book on geometry is an extract from Euclid's Elements, which contains, in addition to definitions, postulates, and axioms, the theorems in the first three books, without proofs. How can this omission of proofs be accounted for? It has been argued by some that Boethius possessed an incomplete Greek copy of