a multiple of the third as the second is of the first, and the second being negative, therefore the fourth must be negative. Let the second be positive and the third negative, that is, let it be 1.a::-b. to a fourth. Now, whereas this fourth must be such a multiple of the third, as the second is of the first, and the second and first being positive and the third negative, the fourth cannot be otherwise than negative. Lastly, let both the second and third be negative, that is, let it be 1.- a :: b. to a fourth. Now the second being here a negative multiple of the first, it follows that the fourth must be a negative multiple of the third. But the third is already negative, and therefore the fourth must be positive. Wherefore the product of a into b will be ab. That of a into b will be ab. That of -a into b will also be — ab. That of a into-b will be ab. That of That of a into - b into c will be - abc; ab, and — ab into c will be- abc. b into c will be abc.'

a into b into c will be abc. because a into b will be And the product of a into

Is it necessary to comment on such an explanation?

In illustrating the principles of the fluxionary calculus, (Vol.II.) the definition of a fluxion is thus given:

Any infinitely little portion of a variable quantity is called its Difference or Fluxion; when it is so small, as that it has to the variable itself a less proportion than any that can be assigned; aud by which the same variable being either increased or diminished, it may still be conceived the same as at first.'

That these differential quantities are real things, and not merely creatures of the imagination, (besides what is manifest concerning them, from the methods of the Ancients, of polygons inscribed and circumscribed,) may be clearly perceived from only considering that the ordinate MN in a digram annexed moves continually approaching towards BC, and finally coincides with it. But it is plain, that, before these two lines coincide, they will have a distance between them, or a difference, which is altogether inassignable, that is, less than any given quantity whatever. In such a position let the lines BC, FE, be supposed to be, and then BF, CD, will be quantities less than any that can be given, and therefore will be inassignable, or differentials, or infinitesi mals, or, finally, fluxions.

Thus, by the common Geometry alone, we are assured that not only these infinitely little quantities, but infinite others of inferior orders, really enter the composition of geometrical extension. If incommensurable quantities exist in Geometry, which are infinites in their kind, as is well known to Geometricians and Analysts, then infinitesimal magnitudes of various orders must necessarily be admit ted.'

How it can happen that a distance between two lines should be inassignable, that is, less than any given quantity whatever, we confess ourselves not able to comprehend.

Our custom of not introducing diagrams precludes us from presenting the reader with the method by which Infinitesimals are proved to exist; and we must proceed:

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Now,

Now, to avoid paralogisms, into which it is but too easy to fall, it will be needful to reflect, that infinitely little lines of any order, (agreeably to what obtains likewise in those that are finite,) have two important circumstances to be considered, which are their magnitude and their position. And as to their magnitude, I think they cannot be rejected except by those, who fancy such infinitesimal quantities to be mere nullities."

May we not be excused the labour of refuting these absurdities?

At p. 18. the fluxion of xy is thus found. When becomes x+x, and y is y+y', xy becomes xy+xy⋅+yx•+x'y': subtract xy, and there remains xy+yx+xy: but xy is the rectangle of two infinitesimals, and therefore is infinitely less, and must be supposed entirely to vanish: therefore, &c.

In other instances, such as in drawing tangents, in finding areas, lengths of curve-lines, &c. where some difficulty occurs in explaining and establishing the principles of the method, the learned author is equally unsatisfactory.

P. 203. a line co is found = a+by; the integral of

a2

this,' says the Signora, after a long calculation, which, to avoid being tedious, I shall omit, will be found to depend on the logarithm, or, which is the same, on the quadrature of the hy perbola.' She therefore resolves the quantity a++b2 y2 _into an infinite series, and finds the fluent of each term. We have noticed this part for two reasons; 1st, on account of the affirmation that the fluent can only be found by a long calculation, and 2dly, because of the praise which Mr. Hellins bestows (Advertisement, p. ix.) on the lady for her great skill. We apprehend that the following process, by which the fluent is to be found, cannot be called either long or tedious; in fact, it is shorter than the method by infinite series, which does not completely resolve the problem:

In fine, to adopt Signora Agnesi's explanation of the fluxion, ary calculus, and to believe in the existence of infinitesimals, would be voluntarily to return to that mental bondage from

which we may be supposed to have escaped: it would be at fond election of evil; an unaccountable love of absurdity; and with respect to knowlege, depth, resource in methods, and ingenuity in artifices, we have authors far surpassing the Italian lady. In order to be a learned mathematician, a man needs not peruse all the treatises which have been published on the same subject: it will suffice if he studies the best. It is not with with science as with poetry. It would be strange for a person to allege that he did not read Milton because he had read Homer: but, if he has studied Euler's Institutions, he needs not rise early and sit up late to pore over Signora Agnesi's Institutions.

Let us not be misunderstood, however: considering the time at which it was produced, this is a respectable performance; and, viewed as the production of a female, and a young female, possessing other accomplishmeuts, it calls forth Yet we sentiments approaching to those of astonishment. must repeat that it is not of such intrinsic value as to deserve a re-publication: it might have served as a scaffold, by which some parts of the edifice of science might have been reared: but the edifice is now built, stands firmly, and no such scaffold is requisite.

We ought not to omit a tribute of praise to the liberality of Baron Maseres; not simply that pecuniary generosity by which scientific men are enabled to publish the results of their investigations, but that mental liberality, which could patronize a book containing such doctrines as the Baron had uniformly combated with so much toil, at so great an expence, and for so many years. We love the spirit of such an act, though we esteem not its effects in the present instance; and we wish it a long continuance of existence, though we regret that it created the useless labour by which these volumes have been prepared for the public eye.

ART. IV. Travels in Spain, in 1797 and 1798. By Frederick Augustus Fischer. With an Appendix on the Method of Travelling in that Country. Translated from the German. 8vo. pp. 403. 75. Boards. Longman and Rees.

WHOEVER travels through a foreign country, although it

may have been visited by men of superior ability and discernment, will be able, if he has any talents for observation on men and manners, to collect sufficient stores to interest and entertain his readers; even if he should fail in point of valuable and important information. On this account, the present translation of the travels of M. Fischer, who himself

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modestly

modestly pleads his inferiority to Bourgoanne, Townsend, and others, may justly be ranked among the amusing and creditable publications of the time.-In the preface, the reader is required to consider this work in the light of practical notes to Bourganne, &c.; in particular, as it relates to the present state of literature and the mode of travelling in Spain.

In our review of Professor Link's travels in Portugal, &c. (see our last Number) we pointed out some parts, in which we had an opportunity of comparing that writer's remarks with those of M. Fischer: whence it will appear that these travellers do not always accord in their opinions and representations.-Between the two, we shall not presume to place ourselves as judges.

The present journal commences from Amsterdam, and we have an account of the events and disasters of the voyage till the vessel arrived at Bourdeaux: but they present nothing striking nor uncommon. The 24th letter contains a communication from the brother of Mons. de Humboldt, who is well known in Germany, on the Basque language. He observes:

"The language of Biscay deserves the particular attention of philologists, though it has hitherto been too much neglected. Yet on even superficially running over the vocabulary of that language, it appears that, setting aside the nouns which were unknown at the first civilization of that country, and which have been successively borrowed from the Romans, the French, and the Spanish, the Basque has a very great number of words peculiarly its own, and all of which have a character truly original both as to their origin and formation. This primitive language, which is underived, not to say unstolen like most of those now spoken in the south of Europe, from the Latin, seems however to have, in common with the Latin, German, and even the Greek, a great number of radical words, which might serve as guides to etymologists, and afford them light in their researches into this ancient and primitive language, from which perhaps have sprung most modern tongues, and of which it still preserves some valuable remains. Even those who would be alarmed at the dryness of so irksome a pursuit would find a pleasure in observing the manner in which the Biscayans compose the signs of their ideas; that people scarcely employing any but complex signs to express ideas which all other languages represent by simple signs, such as sun, moon, &c. It would be an object of infinite curiosity to a philosopher to observe and pursue the analogy, according to which the Biscayans combine certain ideas, so as to form new signs and express their perceptions; and there would doubtless thence arise many very useful observations on the originality and mode of viewing objects exercised by that ingenious people. Nor is the theory of the Basque language destitute of utility as to the history of languages in general, their peculiar differences, and their formation."-

In many parts of this tour, the events of the day are noted down just as they occurred; and often, as may be naturally sup

posed,

posed, they are not very interesting. On the road to Lerma, the method of curing a sick mule will probably be new to the veterinary surgeon:

The next morning, when on the point of setting off, it appeared that my mule had not eaten and was sick. Immediately there was a long consultation among all the arrieros that were about to load their beasts, and most of them were of opinion we must give him rest; but the whole day passed without his recovering. He was washed with hot wine, a dose of physic administered, and a plaster applied; but all without effect. What could be the reason of all this? Nothing more simple; the animal was bewitched.

To break the charm therefore a quantity of images of saints of all kinds, chaplets, and a large tub of holy water were brought, the animal was dragged under a gateway, his head placed toward the church, he was loaded with images and rosaries, while a toothless old woman muttering a whole litany of ave-marias attempted to exorcise him, and they concluded by inundating him with holy water from head to foot. Four hours after the animal began to eat, and the next day was perfectly well. You will easily imagine, that without rashly despising this sacred bath I might at least, according to the religion I was brought up in, admit some doubts. It therefore appeared to me from certain symptoms that the mule had a strangury, and that since cold is useful in that disorder, the water that was thrown over him might possibly accelerate his cure.'

We extract a part of the author's description of Madrid:

The public squares are used throughout Spain as promenades and places of assemblage. The small towns and even the villages are not without such an open space, which is generally in front of the church. It is there the Spaniards recreate themselves after their labours, or enjoy the warmth of the sun in winter, and even those who scarcely ever quit the town regularly resort there. From this you may casily conceive the appearance of such a spot in the centre. of the metropolis.

It has struck eleven, and a troop of officers of the guard with brilliant accoutrements, monks in black cloaks, charming women in veils embroidered with gold holding the arms of their cortejos, and a party-coloured crowd of all kinds wrapped up in their cloaks, pour from every street to read the advertisements and posting bills (noticias sueltas): "To day there will be a sermon and music at the Franciscans; there will be an opera and such and such plays: to-, morrow there will be a bull-fight, or the novena of San Felipo commences: Lost yesterday at the prado a little girl, and this morning a chaplet: Stolen three days ago such and such a jewel; if it has been taken through want, and if the thief will restore it by his confessor, he shall receive a handsome reward: The day after to-mor row will be sold by auction a large crucifix, au image of the Madona, and a nacimiento (or case containing the infant Jesus with the two other persons of the trinity in wood, plaster, &c.). This evening the procession of the rosary will set out about eight o'clock."

• Mean