be proper to the mechanical philosophy, but is very ill adapted to support the folid fabric of the arithmetical science. But let us attend to the arguments of this most learned man, in defence of so strange an affertion, "A cypher, or arithmetical nothing (fays he) is really the bound of every number coming between it and the numbers next following, but not as a part. A cypher being added to, or taken from a number, does neither increase nor diminish it; from it is taken the beginning of computation, while itself is not computed; and it bears a manifeft relation to the principal properties of a geometrical point." But in what manner are we to conceive the nothing which intervenes between any two numbers, to be their term or boundary? For Euclid defines a term to be the extremity of any thing; implying by the extremity, fomething belonging to that of which it is the bound. But how can a cypher, or nothing, in any refpect belong to number, or. Something? For if nothing be a boundary, merely from its intervention, a point existing between any two disjoined lines, though at the greatest distance from each, must be their common boundary, which is evidently abfurd. Befides, what relation does it bear to a point, which is endued with a generative power, by its flux forming the fimple extenfion of a line, and, at the fame time, every where limiting its progreffion, and fubfifting in infinite capacity in its every part? Where are the real and divine properties to be found in an arithmetical nothing, which Proclus, in the following Commentaries, exhibits in a point? And how can computation originate from a mere non-entity?

But a little confideration will convince us, that this Saracen, or Indian cypher, is nothing more than an arbitrary character, invented for the purpofe of facilitating computation. For, fuppofe the letter (a) to be placed in

its ftead, and to fignify, when connected with the mark for unity, ten, or ten multiplied by one; when connected with the mark for two, ten multiplied by two, and fo on. And again, when placed twice after unity, let it express the fecond power of ten, or one hundred, in this manner, aa; when thrice connected, one thoufand, or the third power of ten, and fo on: fhall we fay, in confequence of this, that (a) is the bound of numbers, and the principle of arithmetic? Or, fhall we not rather fay, that it is an arbitrary symbol, like any other algebraic character, having no real connection with numbers, and depending, for its existence and application, entirely on the will of its inventor. But this opinion is too abfurd to need any farther refutation.

7. It may here, perhaps, be expected, that I should explain how, in the language of Syrianus, "divine number proceeds from the immortal retreats of unity, until it arrives at the divine tetrad ;" and that I fhould unfold the properties of the tetractys, according to the Pythagoreans; but an undertaking of this kind, would not only. far exceed the limits of this differtation, but, perhaps, in the prefent age, might be juftly deemed, by the lovers of wisdom, a prostitution and profanation of the most exalted truths. Enough, I hope, has been faid to excite the curiofity, and roufe the attention of the thinking and liberal part of mankind; and those who understand what is here briefly delivered, may apply themselves, with advantage, to Proclus on Plato's Theology, where they will find all the mysteries of numbers unravelled; and to the works of the great Plotinus, who will lead them into the pene

In Ariftot. Meta. p. 113. Gr. vel 59. b. Lat.

For the tetrad contains all numbers within its nature, in the manner of an exemplar; and hence it is, that in monadic numbers, 1, 2, 3, 4, are equal to ten,

tralia 2

tralia of the most recondite wifdom. But, in perufing the works of these great men, the reader muft not expect to find the fublimeft truths explained in a familiar manner, and adapted, like many modern publications, to the meanest capacities. For this, indeed, is impoffible to be effected. "Mankind (fays Petvin *), are not to be made any more truly knowing than happy by another's understanding.There is no man can at once convey light in the higher fubjects, to another man's understanding. It must come into the mind from its own motions, within itfelf: and the grand art of philofophy, is to fet the mind a-going; and, even when we think nothing of it, to assist it in its labour." After which he obferves, that "the ancients never attempt to lead us into knowledge, by a continued chain of reafoning; on the contrary, they write in fuch a manner, as to force us to think for ourselves." And, previous to this, he remarks, "that there are certain truths acquired by a long exercise of reafon, both in particular, and likewife in thofe fubjects that are most general, as much, perhaps, out of the reach of the greatest mathematician, as Sir Ifaac Newton's fpeculations are above the capacity of fome that are now called mathematicians." The truth of this observation is fufficiently evinced, in Plato's definition of a philofopher (in his Sophifta), "The philofopher (fays he) is the man who fufficiently fees one idea every way extended through many, every one of them lying apart; and many ideas different from one another, externally comprehended under one. And farther, one idea, throughout all manys, wrapt up in one; and many ideas, every way feparate or difcrect. This is to have the knowledge to difcern how ideas, as they are general, agrce and difagree." Now, he who thinks that a perception of this kind may be acquired

Notes to Letters on Mind, page 83.

by

In

by barely reading an accurate difcourfe on the nature of ideas, compofed in intelligible terms, without, at the fame time, employing a long course of profound meditation, and patient thought, knows but little the difficulty of the task, and until he changes his opinion will never be the wifer. But the folly and prefumption of men, with respect to this fublime philofophy, is really unpardonable; for there are very few who conceive that much previous inftruction is requifite to its acquifition; but almost every man decides peremptorily on the most abstract speculations, and reckons himself fufficient for the moft profound investigations. the fciences and arts they are willing to proceed to perfection by gradual advances; but they confider philosophy as easy, of instant access, and hastily approach to her embraces with an affured confidence of fuccefs. Though, like unhappy Ixion, through their prefumption, inftead of a goddefs, they grafp nothing but an empty cloud. Plato was fo fenfible of this truth, that, in his seventh epiftle to Dion, he exprefsly affirms, that he neither has written, nor ever will write explicitly concerning thefe fublime fpeculations; "For a thing of this kind (fays he) cannot be expreffed by words, like other difciplines, but by a lafting familiarity, and conjunction of life, with this divine object, a bright light

on

This bright light is no other than that of ideas themselves;, which, when it is once enkind. led, or rather re-kindled in the foul, becomes the general standard, and criterion of truth. He who poffeffes this, is no longer the flave of opinion; puzzled with doubts, and loft in the uncertainties of conjecture. Here the fountain of evidence is alone to be found.-This is the true light, whofe fplendors can alone difpel the darkness of ignorance, and procure for the foul undecaying good, and fubftantial felicity. Of this I am certain, from my own experience; and happy is he who acquires this invaluable treasure. But let the reader beware of mixing the extravagancies of modern enthufiafm with this exalted illumination. For this light is alone brought into the mind by fcience, patient reflection, and unwearied meditation: it is not produced by any violent agitation of fpirits, or exta'y of imagination; for it is far fuperior to the energies of thefe: but it is tranquil and fleady, intellectual and divine. Avicenna, the Arabian, was well acquainted with this light, as is evident from the beautiful defeription he gives of it, in the elegant introduction of Ebn Tophail, to the Life of Hai Ebn Yokdhan. "When a man's defires

VOL. I.

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on a fudden, as it were leaping from a fire, will illuminate the foul, and there preserve and nourish its fplendor. He adds, that a publication of fuch concerns, is alone useful to a few of mankind, who from some small vestiges previously demonftrated, are fufficiently fagacious to their invention. But it will fill others partly with a base contempt, and partly with a rash and vain confidence, as if they had now learned fome very excellent things." He then fubjoins the following instance of the difficulty attending such an undertaking: "There are three things (fays he), from which fcience muft neceffarily be produced; but the fourth is fcience itself. And it is requifite to establish the fifth as that which is the object of knowledge, and has a true existence. One of thefe is the name of a thing; the second its definition; the third the resemblance; the fourth science. Now take each of these, defiring to learn what we have lately afferted, and think concerning them all, in a fimilar manner. A circle is called fomething, whofe name we have juft expreffed. After this follows its definition, compofed from nouns and verbs. For that which every where is equally distant from the extremes to the middle, is the definition of that which we fignify by the name of a round, and a circumference, and

defires (fays he) are confiderably elevated, and he is competently well exercised in these speculations, there will appear to him fome fmall glimmerings of the truth, as it were flashes of lightning, very delightful, which juft fhine upon him, and then become extinct. Then the more he exercifes him felf, the oftener will he perceive them, till at last he will become fo well acquainted with them, that they will occur to him fpontaneously, without any exercife at all; and then as foon as he perceives any thing, he applies himself to the divine effence, fo as to retain fome impreffion of it; then fomething occurs to him on a fudden, whereby he begins to difcern the truth in every thing; till through frequent exercise he at last attains to a perfect tranquillity; and that which used to appear to him only by fits and farts, becomes habitual, and that which was only a glimmering before, a conftant light; and he ob ains a conftant and fteady knowledge." He who defires to know more concerning this, and a ftill brighter light, that arifing from an union with the fupreme, muft confult the eighth book of Plotinus' fifth Ennead, and the 7th and 9th of the fixth, and his book on the Beautiful, of which I have published a tranflation.

a circle.