Page images
PDF
EPUB

ners, and language of Transylvania, acknowledge a second country, and a government less subject to such fatal revolutions.' P. 268.

The Walachians are not temperate; yet their manners are simple, and their customs chiefly Turkish. They are idle, covetous, and blindly superstitious.

The embassy soon enters Moldavia; but, in the narrative, the passage of the Danube is strangely omitted; and indeed many circumstances lead us to believe that this is an abstract of a larger and more important work. The Danube is highly useful, both to Moldavia and Walachia; and, with the Dniester on its north-west, renders these provinces best adapted for very extensive commercial enterprises. The neighbourhood of the Black Sea, its connexion with the Mediterranean and with countries the most distant, by its central situation and the course of its chief river, together with the fertility of the soil, might render the inhabitants a very rich and powerful, and, would their government permit, a happy people.

The colonies that would be sent there would have no cause for apprehending the same inconveniences and misfortunes as have been ex. perienced by those of Astracan, because they would be removed to a shorter distance, and would have all the resources of civilised Europe to hope for. They might likewise avoid the inconveniences to which the establishments of the Bannat of Temeswar have been subjected, by being more judicious in the choice of the lands to be fixed on for their habitation. In this view the tracts of Walachia and Moldavia on the Danube are the most favourable, and the climate itself the most salutary of any to be met with. Nothing more would be necessary than to drain the lands, and to carry off the stagnant waters, in order to purify the atmosphere, and render the soil more proper for cultivation. The opening the mines and clearing the woods, the tilling the grounds, and cultivating vines and fruit-trees in a more skilful manner, would be objects which in the space of a few years might enrich two hundred thousand indigent families, who are at present condemned to idleness and want, and bring into the coffers of the sovereign more than sixty millions of livres. The nature of the soil of the plains and hills exhibits in general such favourable properties, that plantations might almost any where and indiscriminately be formed of rice, tobacco, or sugar-productions that are foreign to our continent, and singularly calculated to succeed in this soil. In this corner of Europe would then be collected almost every object of cultivation known in the globe. The desert, which extends from Jassy to the Dniester, and to the frontier of Podlakia, in a space of twenty leagues in breadth and thirty in length, offers one of the best soils that it is possible to meet with for the cultivation of barley, wheat, and orchards. There is not a single tree in all this space; but the land is covered with high verdant grass, which every where announces the abundance of productive salts with which it is impregnated. This land is undulated on all sides by an infinity of small hills, with springs of water at every CRIT. REV. Vol. 38. June, 1803.

1

step. Nothing could be easier than to plant orchards in it, or everr woods, either of which would succeed extremely well.' r. 284.

The account of the Moldavian dances is amusing, but too long for insertion. The ancient history of this province has still less claim to our particular attention; and its present state, its productions, government, &c. contain no facts of very particular importance. Three glasses of its wine, it is said, will intoxicate the hardest drinker; yet, it is added, the liquor is not heady. Some mistake must have occurred in the translation.

The account of the government of Moldavia we must omit. The Moldavians themselves are described as proud, audacious, and quarrelsome; but easily appeased, lively and jocose. Their arins are the bow and the javelin; and the Turk and the Tartar, the Armenian and the Jew, meet no quarter. Moderation is a virtue unknown, either in wine or in other circumstances. They are haughty in prosperity, cowardly in adversity; eager to attempt, but, if foiled at first, immediately discouraged. The men are robust and well made, and have a considerable facility in learning the inilitary exercises.

The Walachians are more lively, have more intellect and courage, but drink like the Moldavians, and are equally, perhaps more, hospitable. The women are handsome, but pale; and their dress displays very accurately their forms. In both provinces, in a circumference of near six hundred leagues, the inhabitants do not exceed sixty thousand, of very different races. Moldavia, we are told, could once furnish forty thousand fighting men. At present, the hospodar could not bring ten thousand into the field. His ordinary revenues amount to about three millions of livres; those of the prince of Walachia to nearly twice as much.

The remainder of the tour is entertaining, but not highly interesting; and, as our article is already sufficiently extended, we shall refer the reader to the work itself, which is not, on the whole, uninteresting or unimportant, though unequal.

ART. VI.-The Principles of Analytical Calculation. By Robert Woodhouse, A. M. F. R. S. &c. 4to. 83. Boards. Rivingtons. 1803.

QUANTITY is of two kinds, continued and discrete. The ancients were more conversant with the former, the moderns, with greater reason, apply themselves to the latter. On the advantages and disadvantages of each, much dispute

has arisen; and a preference has been capriciously given to one or other of the systems, without knowing the grounds on which it should be established. The ancients were compelled by necessity to pursue long and laborious investigations of continued quantity, because their symbols for discrete quantity were only to be managed with extreme difficulty: they made but little progress in arithmetic; and algebra was scarcely known to them. The slight circumstance of the introduction of the Arabian or Indian numerical figures, wrought a prodigious revolution in science: discrete quantity became the object of investigation; algebraic calculations were pursued to a great extent. The doctrine of fluxions, or the differential calculus, was invented; and problems, which would have tortured an ancient geometrician during his whole life, are now solved with great ease by a young mathematician in the commencement of his scientific career.

Mathematics, being the science of quantity, are naturally, from this division of quantity, arranged into two branches; that of continued, and that of discrete quantity; each of which should be studied separately; and, when the student has obtained clear notions of each in his mind, he will readily see the connexion between them, and that they may be made to impart mutual assistance. Thus, having obtained from Euclid the information that the square of the ordinate, in a semicircle, is equal to the rectangle under the abscissas, he can convert this into an algebraic proposition, by making ar-r-y: he now considers his right line as se parated into parts, on which he erects perpendiculars; and, by numbers, assigns to each ordinate its proper magnitude. Thence he ascertains that the greatest square is that of the ordinate which passes through the centre; and the same truth is now established by both algebra and geometry. The difficulty of finding areas is, by the ancient methods, almost insuperable. An unfortunate term introduced by sir Isaac Newton throws an unnecessary ambiguity over the modern system. He changed the meaning of equality, a term of which we have the clearest idea. Throughout all his work, where the words ultimately equal are used, the proper terms should be never equal; that is, the quantities which he affirms to be ultimately equal, are never equal to each other; but their difference is small; and they are approaching to a limit, which limit may be used in his reasonings. Thus, in his second Lemma, the discrete quantities, the interior parallelograms, can never be equal to the area of the curvilinear figure; in the seventh, the chord and tangent, that is, the hypothenuse and side of a right-angled triangle, are said to be ultimately equal, though it is well known by Euclid that

they can never be equal to each other. Yet Newton, in his future reasonings, finding that the limits to which these quantities are approaching would answer his purpose, expresses his intentions by the quaint and incongruous term, ultimately equal.

It was natural that this innovation should excite, among the lovers of perspicuity, a great deal of disgust; and this was increased, when they found that it led them to what they conceived to be a still more perplexed subject-the doctrine of fluxions. The subtle notion of velocity was introduced into every question: quantity was divided into minute parts; and some were rejected, others retained, apparently to those not acquainted with the new doctrines, at random. Hence the whole of this new science appeared, to a very amiable and intelligible writer, a complete tissue of sophistry; and he attacked it with a vigour which the mathematicians in vain endeavoured to elude. They were, however, convinced that the conclusions obtained by their new science were accurate; and, little anxious about first principles, they left the good bishop Berkely in possession of his post, and pursued their investigations, though their defects, as our author justly observes, had in metaphysics and logic been clearly made out.' Some, however, since sir Isaac's time, have endeavoured to give that perspicuity to the doctrine of fluxions, which would have been necessary to satisfy the rigid notions of the ancient geometricians: the attempt does them credit; and their failure ought not to deter others; for mental satisfaction and improvement are more worthy objects, than simple rules and most compendious processes' in computation.

The intention, then, of our author is to conduct us, on the principles of true reasoning, through our analytic calculations; to separate what is strictly true, from mere articles of convention; and to show us where the mind can rest with satisfaction in the discoveries of the moderns. This task he has performed with much ingenuity; but his technical language, together with the novelty and abstruseness of his discussions, will deter numbers from pursuing the chain of his reasonings; and they will rather acquiesce in the ipse dixit of Newton, than go through a laborious mental investigation, which may shake their faith in their former systems. We will point out a few instances in which the author detects the fallacy of modern reasonings.

The nature of positive and negative quantities is the first object that meets us in this work, arising from the explanation of the algebraic signs. The latter are allowed to have a place; but not because any proof can be given of the effects of addition, &c. upon them, but for the sake of com

modiousness in calculation;' and all the rules relative to them are partly arbitrary, and suppose some previous convention.' On this subject he is as much at variance with those who support as those who reject these quantities: the former will not be pleased that he has destroyed all their proofs; the latter will think him inconsistent with himself, and retort upon him the passage we have just quoted, that mental satisfaction and improvement are more worthy objects, than simple rules and most compendious processes in computation.' On this arduous subject, however, we will leave the author to speak for himself.

It has been already observed that, if negative quantities are made the object of demonstration, it must be in consequence of some arbitrary rule. The rule for transposition introduces negative quanti. ties, and leads to equations of no direct meaning; the rule for the multiplication of signs likewise introduces them, for although in all real questions, it can be proved that the rule leads to true results, and therefore, for commodiousness is made general; yet, in the reverse operation of extraction, the consequence of such a rule may be equations, considered separately from questions to which they belong, of no direct meaning; thus if x2, x must be put ±a, since according to the rule a × —a gives a1 as well as tax +a.

The notions that have been sometimes formed of negative quantities are as faulty, as the methods by which rules for their multiplication have been proved; they have been considered as quantities less than, or their nature has been attempted to be explained, by illustration drawn from mercantile transactions, or from the properties of geometrical figures; the first notion is manifestly an absurd one, the second merely illustrative and proves nothing; besides, it in some sort begs the question, for, that, a positive quantity representing a line drawn in one direction, a negative quantity must be used to denote a line drawn in an opposite direction, is by no means a self-evident truth, and is in fact a consequence of the definition, by which the application of algebra to geometry is made, and of certain properties of negative symbols previously established by the rules of algebra.

The method of proving the rule for the multiplication of signs, by multiplying (a-a) into b contains a manifest fallacy, for it is essential to the proof, by which (a-a) ±b is made equal to ± ab F b2, that, a—a should be a positive quantity: it cannot at once be put =0 and employed in reasoning as a quantity; or if first considered as a quantity and afterwards puto, the hypothesis is shifted, and the previous result must be abandoned.

-a

The method of proving the rule, by reasoning on the signification of the word subtraction, is not more satisfactory; for, to subtract a negative quantity, does not necessarily mean to add a positive one; if the phrases are equivalent, they must be made so by definition; there is a wide difference, between what is agreeable to the analogy of language, and what is admissible in strict demonstration.' P. 7.

The author makes a just distinction on the meaning of the sign, which is confounded by most mathematicians.

« PreviousContinue »