to take the word nearly, though not exactly, in what seems to have been its most usual sense; not indeed for comforters or cherishers, but for those who promote comfort and convenience, viz., ministers or servants. It does not at all follow, because he is blaming the introduction of these persons as expensive, superfluous, and otherwise evil, that he describes them by a word expressive of evil. As a ministering angel would be a reheting angel, so I take a rehetor here to be simply a minister, one who waits upon your occasions and serves you.

A.N.

ARABIC NUMERALS.

The history of the Arabic numerals, as they are generally called, is so mixed up with that of the use of the decimal scale, that they form, in fact, but a single inquiry. The mere history of the bare forms of symbols has, doubtless, its use: but then it is only in the character of matériel for a philosophical discussion of the question—a discussion into which the natural progress of the human mind and the urgency of social wants must enter largely.

It might at first sight appear, from the cognate character of the Hebrew and Arabic languages, that the idea of using a single symbol for each number, might originate with either—with one as likely as with the other. But on reflection it will readily appear that the question rather resolves itself into one respecting the "hand-cursive" of the Jews and Saracens, than into one respecting the constitution of the languages. Of the Jewish we know nothing, or next to nothing, at the period in question; whilst the Arabic is as well known as even our own present style of calligraphy. It deserves to be more carefully inquired into than has yet been done, whether the invention of contracting the written compound symbols of the digital numbers into single symbols did not really originate amongst the Jews rather than the Saracens; and even whether the Arabs themselves did not obtain them from the "Jew merchants" of the earlier ages of our era. One thing is tolerably certain:—that the Jew merchant would, as a matter of precaution, keep all his accounts in some secret notation, or in cipher. Whether this should be a modified form of the Hebrew notation, or of the Latin, must in a great degree depend upon the amount of literary acquirement common amongst that people at the time.

Assuming that the Jews, as a literate people, were upon a par with their Christian contemporaries, and that their knowledge was mainly confined to mere commercial notation, an anonymous writer has shown how the modifications of form could be naturally made, in vol. ii. of the Bath and Bristol Magazine, pp. 393-412.; the motto being valent quanti valet, as well as the title professing it to be wholly "conjectural." Some of the speculations in it may, however, deserve further considerations than they have yet received.¹

The contraction of the compound symbols for the first nine digits into single "figures," enabled the computer to dispense with the manual labour of the abacus, whilst in his graphic notation he retained its essential principle of place. It seems to be almost invariably forgotten by writers on the subject, what, without this principle, no improvement in mere notation would have been of material use in arithmetic; and on the other hand that the main difference between the arithmetic of the abacus and the arithmetic of the slate, consists in the inevitable consequences of the denotation of the single digits by single symbols.

The abacus, however, in its ordinary form, is essentially a decimal instrument: but its form was also varied for commercial purposes, perhaps in different ways. I never heard of the existence of one in any collection: but there is preserved in the British Museum a picture of one. This was printed by Mr. Halliwell in his Rara Mathematica—not a fac-simile, but a rule and type representation of it, ciphers being used by him for the circles in the original. Mr. Halliwell gives it without note or remark; and evidently had not divined its meaning. This was done, however, soon after in a review of Mr. Halliwell's book in the Philosophical Magazine. I am not able at this moment to refer to either, so as to give exact dates: but is was somewhere from 1838 to 1840.

Perhaps, however, I am giving "E.V." information that may be irrelevant to his purpose; though it may of some use to another class of inquirers. I proceed, therefore, to one or two notices that seem to have a more direct bearing on his object:

1. Chasles' Aperçu Historique sur l'Origine et le Développement de Méthodes en Géométrie; passim, but especially in note xii.: 4to., Bruxelles, 1837.

2. Chasles' several notices in Comptes Rendus des Séances de l'Acad. des Sciences. All subsequent to the "Aperçu."

His Catalogue des Manuscrits de la Bibliothèque de Ville de Chartres should also be consulted, if accessible to "E.V." Copies of it, however, are very rare in the country, as it was privately printed and never published. If, however, your correspondent have any serious inquiry in view which should render his consultation of it desirable, I can put it in his power to do so personally through you.

3. Libri, several notices in the same series of papers.

4. Libri, Histoire des Sciences Mathématiques en Italie. Several places. Bactulica. Paris, 1838-1841. 4 tomes. 8vo.

5. Peacock (Dean of Ely), "Arithmetic," in the Encyclopædia Metropolitana. This is now, I believe to be had either separately, or in the volume devoted to pure "Mathematics."

6. De Morgan, Penny Cyclopædia in loc., and occasionally elsewhere in the work.

7. Leslie's Philosophy of Arithmetic.

8. Humboldt, in a paper which is translated in the Journal of the Royal Institution, vol. xxix.

I believe a good many other references might be made, with little trouble, to foreign Mémoires; and (perhaps still more to your correspondent's apparent purpose) to some amongst the Mémoires that relate to inscriptions and topography, rather that amongst those relating directly to science or literature. However, the two parts of the subject cannot be effectively studied separately from each other; and I am not without a hope that these straggling notes may be of some use to "E.V."

Under the view of inscriptions it occurs to my memory that in two or three places on the church of St. Brelade in Jersey, there are marked four vertical straight lines, which are interpreted by the natives to signify the Arabic numerals 1111; as the date MCXI of the building of the church. The church is evidently a very ancient one, and it is agreed to be the oldest in the island, and the island historians assign it to the early part of the 12th century. For these symbols being coeval with the building I do not vouch: as (though it is difficult to say what may constitute antiquity in the look of four parallel lines) I confess that to my eye they had "as modern a look" as four such lines could well have. The sudden illness of one of my party during our visit (1847), however, precluded my examining that beautiful spot and its interesting little church with the care I should have wished.

I may be allowed to suggest the necessity of some degree of caution in discussing this question: especially not to assume that any Arabic numerals which appear in ecclesiastical inscriptions are coeval with the dates they express; but rather inquire whether, from the condition of the stone bearing the inscription, these numbers may not have been put there at a later period, during repairs and alterations of the building itself. It is for many reasons improbable, rather than otherwise, that the Arabic numerals should have