applied with any degree of success either to civil affairs or to scientific computations. The number of the present year, 1830, for example, would require, according to this method, no less

than twelve marks or characters, ₹99999999nnn,

to express it, or three times the number of signs required by the Arabic notation. At the same time, the high antiquity of this method of numeration is evinced by the simplicity of the principle upon which the scale is constructed, no less than by the age of the monuments in the inscriptions on which it is discovered. It may be added that this scale has no peculiar or distinct set of numerical signs appropriated to the days of the month, as is the case both in the hieratic and enchorial forms of notation.

The hieratic form, which is the most complete of all, possesses several very remarkable peculiarities; but as it passes naturally into the enchorial, and has a much more marked affinity to that form than to the hieroglyphic, of which some explanation has been just given, we shall confine ourselves at present to a mere exposition of the principle of the scale; reserving for the account of the last or civil form of numeration such details as are deemed necessary for expounding and illustrating the mode in which the numerical signs were discovered and ascertained. We may observe in the outset, however, that the principle of the hieratic is essentially the same with that of the hieroglyphic scale, only it is carried much farther, and a greater variety of signs admitted, which renders it capable of indefinite extension. The digits, omitting the variations, which are in general trifling, are represented thus: 1, or 7; 2, 4; 3, 4; 4, uy; 5, 7 or

7 ; 6, ≤ ; 7, ₪ or λ ; 8, ⇒ ; 9, 2 or 3.

Ten is

; or 20

represented by lambda either direct or reversed, or byor; 30 by; 40 by; 50 by; 60, by

14 or 1; 70 by ; 80 by

; and 90 by 1. The

of 300, of 400 ";

while 500, 600, 700, 800, and 900 are represented respectively by combining the signs of 200 and 300, of 300 and 300, of 300 and 400, of 400 and 400, and 300 thrice repeated. The mark for 1000 is just the sanpi of the Greeks, viz. b or 5 or b

that for 2000, for 3000, for 4000; 5000, 6000, 7000, 8000, and 9000 being represented by precisely the same arrangements as 500, 600, 700, 800, and 900. The symbol of 10,000 is 7, and 100,000, is represented by the sign of 100 combined with that of 1000, thus 5. So much for the numeri vulgares of the hieratic scale. With respect to the numeri dierum, they are not a little remarkable as exhibiting the source whence the Saracens derived three if not four, of the numerals which that victorious people afterwards introduced into the western world, thus conferring upon Europe one of the greatest obligations it ever received, at the hand either of conqueror or sage, the art of printing alone excepted. These numbers resolve themselves into three decades, the first of which is as follows, viz. 1, J ; 2,2 or 3 ;

3, 3 ; 4, 7 ; 5, 23 ; 6, 33 ; 7, 37 ; 8, nn ; 9,

10 or. The numbers composing the second decade, or from 10 to 20, are represented by combining the symbol of 10 with the digits in succession, thus 11; 2/ 12; 3/ 13; 14, and so on to twenty, the sign or mark of which is. Lastly, from 20 to 30 (the symbol of which is

or

the numbers are represented in the same way precisely as from 10 to 20, viz.,, 21; 21, 22 ; 31, 23, &c. So much, then, for the hieratic notation in both its parts, which is evidently, in many respects, a great improvement upon the hieroglyphic, the source whence it was primarily derived.

The enchorial form of notation, is still unfortunately the most imperfect of all, at least in the numeri vulgares, in which there is a hiatus from 13 to 20, and from 60 to 100, the intervening numbers being as yet undetermined by actual discovery, although easily conjecturable from the analogy of the scale. It may not be improper, therefore, to explain to the reader how those signs which are already ascertained

came at first to be discovered, as well as to point out a few of the remarkable verifications which these discoveries have received, in consequence of the fortunate accidents which have so materially contributed to the advancement of Egyptian learning. And here it may be stated, that several of the enchorial signs of numbers have been collected from the Rosetta Inscription, and also from the exordia of those enchorial papyri in which a registry in Greek happens to be adscribed to the Egyptian syngraph or deed. The registry, it will be observed, invariably specifies the amount of the tax which, by the law of Egypt, was imposed on every transfer of property; and as this tax appears to have been for the most part paid in the course of the year in which the transfer was effected, it follows that the year mentioned in the registry must in general be identical with that specified in the commencement of the relative enchorial syngraph or deed. Thus, in the 38th Berlin papyrus, the first words of the Greek registry are rovç λá, while the enchorial syngraph begins 14, 31, 1; anno, with Egyptian words to precisely the same effect: so that, from the exordium of this syngraph, we ascertain the enchorial mark or sign for the number 31, viz. 14. But all the enchorial numerals cannot be discovered in this way; for although dates are entered in the syngraphs as well as in the registries, yet the date of the purchase, engrossed in the syngraph, is often different from that of the payment of the tax, specified in the registry. Thus, in the 36th Berlin papyrus, the date of the purchase is the 18th Athyr (??, 18, LD , Athyr), whereas that of the payment of the tax is the 9th Choiak (xoix d′), the month immediately following: Hence it is obvious that the Greek numbers of days, entered in the registries, could go but a short way in enabling us to determine the enchorial numbers of days, as recorded in the syngraphs; nor was the difficulty lessened by the circumstance of the enchorial signs, indicating the numbers of days, being arranged upon a principle totally different from that of the signs applicable to the numeration of years and other matters indiscriminately. Without some additional means therefore, in the shape of monuments, affording opportunities of comparison and verification, we might still have remained in nearly total ignorance of the numeral system of the ancient Egyptians; for although the import of certain marks or signs

might have been conjectured upon the principle above mentioned, nevertheless the frequent discrepancy between the dates of the registries and those of the syngraphs, coupled with the want of any clue by which their identity or difference could be ascertained, must have rendered it utterly impossible to determine the value of a single sign with any thing approaching to certainty. Fortunately, however, new monuments were discovered, and all these difficulties overcome. "Au milieu des inappréciables richesses archæologiques réunies dans la ville de Turin par la mémorable munificence de S. M. le Roi de Sardaigne (says the Bulletin Universel for May 1825), M. Champollion signala publiquement à l' Europe, dès le mois d' Octobre 1824, les nombres historiques qu'il avoit reconnu le premier, et donna dès ce moment un certain nombre de dates historiques tirées des chiffres de ces manuscrits. Il s'occupa dès-lors à compléter le tableau des chiffres hiératiques et démotiques ; et des dessins qu'il reçut en même temps par un heureux hazard, de M. Anastasy et de M. Salt, consuls de Suède et d'Angleterré en Egypte, lui ayant fourni les élémens qui pouvoient lui manquer, il fut en état, dès le mois de Novembre 1824, de présenter au monde savant le tableau complet des chiffres et du système numérique des Egyptiens." This he accordingly did, in a criticism on an Italian work published at Turin under the title of Saggio sopra il Sistema de' numeri presso gli antichi Egiziani; and it is but justice to add, that few additions of any consequence have since been made to the "tableau des chiffres et du système numérique des Egyptiens" which M. Champollion then presented to the world.

So much for the fact of the discovery itself. With regard to the mode in which it was effected, à circumstance about which the inquisitive reader may naturally be supposed to feel some curiosity, nothing can possibly be imagined less intricate or mysterious, so far as the mere principle is concerned. The frequent discrepancies, in the matter of dates, between the Greek text of the registries, and the Egyptian text of the syngraphs, have been already noticed; and we have also explained the cause to which these descrepancies were owing, as well as the obstacles thus interposed to the certain determination of the Egyptian signs of numbers contained in the text of the syngraphs. But it is evident that all these obstacles would be at once removed, and that every difficulty would vanish, provided we were fortunate enough either to obtain bilingual syngraphs, or, which comes to the same thing, to discover separate translations of any of those previously known to us. Now, it so happened, by a rare concurrence of chances, that both these objects, so vitally important to the interests of Egyptian learn

ing, were much about the same time realized; that in the great Turin papyrus, containing an account of the litigation between Hermias and Horus, M. Peyron discovered a translation of the first enchorial syngraph of sir George Grey, relative to the purchase of a piece of ground near Diospolis Magna by one Teephbis; and that several bilingual manuscripts, in which dates and other numbers occurred, were also brought to light. In a word, when human sagacity was utterly at fault, and the cultivators of Egyptian literature and antiquities au desespoir, fortune supplied the key which was wanting, and thus laid open a new compartment in the colossal fabric of which a small portion only had been previously explored. With these appliances and means to boot, M. Champollion had a comparatively easy task in evolving the numerical system of the antient Egyptians.

It only remains to subjoin an example or two in illustration of this general statement, which might otherwise prove scarcely intelligible to those unacquainted with the history and progress of discovery in this new and interesting field of inquiry. And here we shall recur to the remarkable syngraph of Grey, above mentioned, and of which a translation was so unexpectedly found. It bears that a piece of ground to the south of Diospolis had been purchased by one Teephbis; and the registry informs us that he completed his purchase on the 18th day of the month Pachon, and in the 28th year of a certain king's reign: ὅν ἠγορ ̓ ἐν τῷ κή παχῶν ιή. But the translation or summary of the deed, engrossed in the great papyrus of Turin, contains precisely the same date, and consequently establishes the identity of the date in the syngraph with that in the registry. All that remains, therefore, is to distinguish the enchorial groups in the syngraph corresponding to the numbers 18 and 28: and this is easily done; for the characters representing Pachon and year being previously known, the loci of the groups answering to the numbers, as well as the groups themselves, are thus fixed and ascertained; and this is rendered still more certain by the circumstance of the signs or characters representing the number 18, having been determined anteriorly by an examination and comparison of other manuscripts. Again, in the Turin papyrus, the extent of the ground purchased by Teephbis is given; he bought, it seems, seven and a half house-cubits: ηxes Oikoπεδικοὺς ἑπτὰ ἥμισυ : and this number occurs four times in the translation or summary. On turning to the enchorial syngraph, we accordingly find, in combination with the group answering to cubits or house cubits, two other groups of characters easily distinguishable, which occur in lines 12, 13, 15, and 18, that is, four several times, and which therefore correspond to the num