The Hebrews, Greeks, and Romans represented numbers by the letters of the, alphabet peculiar to each nation. The most simple method of notation among the Greeks was, by making their 24 letters represent each a number in order, from 1 to 24; this method may be seen by referring to Homer's Iliad, or Xenophon's Cyropædia, where it is employed in numbering the books: higher numbers were represented by small letters pointed underneath, by the capitals, and by inclosing the capitals with the Greek II, &c.

From Greece Arithmetic passed to the Romans, who do not seem to have made any improvement in the science; they merely adapted the letters of their alphabet to the numbers received from their masters. Their method, which is still employed in denoting dates, chapters, and sections in books, ought to be understood by every one, and is as follows. I stands for one, V five, X ten, L fifty, C one hundred, D five hundred, M one thousandf; these seven letters, differently placed or marked, were made to express all numbers. As often as any character is repeated, so many times its value is repeated; thus, II represents two, III three, XX twenty, XXX thirty, CC two hundred, MM two thousand. A less character placed on the left of a greater diminishes its value; thus, IV denotes four, IX nine, XL forty, XC ninety. A less character to the right of a greater increases its value; thus, VI denotes sir, VII seven, XI eleven, LX sixty,

f The derivation of these numerals is thus given by some: I, denoting initium, the beginning, was considered as the only fit representative of the first number, or one. V, (the ancient U,) being the fifth vowel, was with propriety put for five. X, being made up of two V's, represented two fives, or ten. C, centum, or one hundred. M, mille, or one thousand. L, being the half of the old C, which was square, was put for half a hundred, or fifty. D, dimidium mille, or half a thousand, five hundred. The D was frequently written IƆ, and the M, CIƆ; hence these latter marks are sometimes put for 500 and 1000 respectively.

CXX one hundred and twenty, DX five hundred and ten, DCC seven hundred, M,DCCCC,XC,IX one thousand nine hundred and ninety-nine. In some ancient books, records, and inscriptions, and on antique coins and medals, we meet with the C inverted; thus, IƆ denotes five hundred: every added increases it tenfold; thus, 1ƆƆ denotes five thousand; CIO stands for one thousand, and a C and added at the ends increase its value tenfold ; thus, CCOO denotes ten thousand, CCCIɔɔ one hundred thousand, cccciɔɔɔɔ one million; a line over any number increases its value a thousand-fold; thus, VIII denotes eight thousand, X ten thousand, LXXX eighty thousand, CC two hundred thousand, MMM three million, &c.

We have not the means of tracing the progressive improvements of Arithmetic among the ancients; judging from their works, (which however are not always to be depended on,) there is reason to suppose that the science advanced. Beside Addition, Subtraction, Multiplication, and Division, the ancients possessed methods of extracting the Square and Cube Roots; they were acquainted with the theory of Proportions; Arithmetical and Geometrical Progression; and in general with the combinations of numbers, the reduction of ratios to their simplest form, &c.

The ancient methods of notation were, however, but ill adapted to the practical operations of Arithmetic; and hence it is that the art, with respect to its practical part, must have made but slow progress. The destruction of

Although the greater part of heathen antiquity has descended to us through the hands of the Greeks, yet their evidence must be received with caution, particularly that of the Helladians; they were a bigotted people, highly prejudiced in their own favour. There surely was never any nation so incurious and indifferent about the truth. Bryant's Analysis, vol, i, p. 143. 155.

the famous Alexandrian library, A. D. 642, has left us no particular treatise on the subject; we have, however, some of the most plain and useful properties of numbers in the seventh, eighth, ninth, and tenth books of Euclid's Elements, A. C. 280, and in the Arenarius of Archimedes, A. C. 220; there is likewise the Commentary of Eutocius on Archimedes' Treatise of the Circle, some fragments of Pappus, A. D. 400. The writings of Nicomachus, A. D. 100, which were published at Paris in 1538, and the treatise of Boethius', written at Rome in the sixth century, give us no very favourable idea of the ancient Arithmetic, which seems to have consisted principally of dry and tedious distinctions and divisions of numbers; so that on the whole, the acquisition of any considerable degree of knowledge in this most useful branch must have been attended with almost insurmountable difficulties*.

Pappus was an eminent mathematician and philosopher of Alexandria; he lived in the fourth century after Christ; the greater part of his valuable writings are lost. His Mathematicul Collections, in eight books, except the first, and part of the second, are still extant; parts of these have been published by the following authors, viz. Commandine, in a Latin translation with a commentary, 1558; Mersenne, 1664; Meibomius, 1655; Wallis, 1688; David Gregory, 1703; and Dr. Halley, 1706; also Dr. Hutton has given a brief analysis of these books in his Mathematical Dictionary, p. 187, 188. vol. ii.

i Boethius was a celebrated Roman; he was put to death, A. D. 525, by Theodoric, king of the Ostrogoths, on suspicion of a conspiracy. During his confinement he wrote that excellent work De consolatione Philosophia. The best editions of his works are that of Hagenau, 4to, 1491, and that of Leyden, cum notis variorum, 1671.

Aldhelm, bishop of Shireburn, and one of the most learned men of the age, who flourished in the time of the Saxon Heptarchy, A. D. 700, complains bitterly of the difficulties he met with in learning Arithmetic, as almost surpassing the powers of the human mind. He thus writes to his friend Hedda, bishop of Winchester. "What shall I say of Arithmetic, whose long and intricate calculations are sufficient to overwhelm the mind, and throw it into despair? All the labour of my former studies, by which I made myself a complete master of several sciences, was trifling in comparison of what this cost me." Anglia Sacra, t. ii. p. 6, 7. quoted by Dr. Henry.

Psellus, who lived in the ninth century, wrote a compendium of the ancient Arithmetic in Greek, which was published by Xylander, A. D. 1556, in Latin; and a similar work was written shortly after in the same language by Iodocus Willichius. These works are at present objects rather of learned curiosity than use; few persons will take the trouble to understand them.

The Arabs, who had shewn themselves the most inveterate enemies of learning, by a revolution of sentiments not uncommon, became its most zealous supporters. From them Arithmetic received some of its most useful improvements; among which the method of notation at present in use may be considered as the chief. It does not appear, however, that the Arabians ever laid claim to the invention; they refer us to the Indians; and hence the figures employed in our calculations are sometimes called Indian characters.

The Arabs were in possession of the Indian method of notation probably for the space of three centuries before the Europeans knew any thing of the matter. The latter were involved in the darkest ignorance, which the genius and learning of the few great men this age of blindness produced were unable to dispel, and which served only to render that mental darkness visible in all its horrors.

Among the few illustrious characters which appeared at this period, Gerbert' deserves the first place. This

1 Gerbert was born of mean parents, but it is uncertain in what year. Having spent several years among the Saracens at Corduba, during which he industriously collected all that was valuable of their Geometry, Astronomy, and Arithmetic, he returned to France in 970, where he was caressed by the wiser part of his countrymen; but the generality of them treated him as a redoubtable Magician: and the credulous writers of those times relate many ridiculous stories about him; as that he understood the language of birds; that he could raise the Devil, was very familiar with him, and bequeathed his soul to him after death, &c. &c. See Vincent's Lectures against Popery, Lect. VII. p. 191. a book in which many stories of the kind are to be found.

literary hero possessed an enlargement of mind, and a thirst for knowledge, rarely to be met with. He was educated in the monastery of Fleury; but soon discovering the incapacity of his teachers, he fled from his monastery, and went to Spain, which was then under the dominion of the Arabs. Having fixed himself at Corduba, he applied with ardour to the acquisition of the Arabian language, and the sciences which that people almost exclusively possessed; he succeeded so well, that in a few years he returned to France, and enriched the Christian world with the literary spoils obtained from the Mahometans, A. D. 960. To him the nations of Europe are indebted for the most valuable of all his acquisitions, a knowledge of the Arabian numeral figures, on the use of which depends every subsequent improvement in Arithmetic.

The Arabian method of notation introduced from Spain by Gerbert, notwithstanding its advantages, was not so eagerly adopted as one might be led to expect; 150 years having elapsed before it was known in Britain, and nearly 100 more before it was brought into common use, as is shewn by Dr. Wallis.

The first writer of note after the reception of the Arabian method, was Jordanus of Namur in Flanders, about the year 1200; his work was commented on, and published shortly after the invention of printing, by Johannes Faber Stapulensis, viz. in 1480. Johannes de Sacro Bosco, an Englishman, wrote a treatise on Arithmetic in the thirteenth century; as did Maximus

Gerbert was preceptor to Robert I. King of France, and to Otho III. Emperor of Germany. He was Bishop of Rheims, and afterwards Archbishop of Ravenna. At length, on the death of Pope Gregory V. A. D. 998, Gerbert was, by the influence of his pupil Otho, chosen to succeed him on the Papal throne, under the name of Sylvester II. He died about the year 1003.