Euclid existed nearly three hundred years before Christ, and that [he is] yet nearly a stranger to those Elements which have conferred imperishable renown on their author and compiler." Mr. Swale was ever anxious that the ancient geometry should be in the ascendant, and to this end he never omitted an opportunity of impressing the beauties of his "Divine Geometry" upon the minds of his junior correspondents. A few statical and dynamical problems are inserted in this volume, but they present no difficulties worthy of particular notice. The most important enquiry, perhaps, is that which determines the direction of impulsion of a billiard ball on a triangular table, “so that it may for ever pursue the same track," to be the sides of the triangle of minimum perimeter inscribed in the given triangle.

Volume xiv. is a bulky octavo, bearing the same title as the preceding. It opens with a series of "Lessons for his Son," amongst which are no fewer than fourteen original methods of dividing a given line in extreme and mean ratio. Several of the isolated solutions to other questions contain references to the Geometrical Amusements, and were probably intended for the succeeding portions of that valuable work. The most important portion of the volume, however, is a short discussion of the different cases of the problem," to determine P in a line MN of any order, so that drawing the tangents PV, PT, to two given circles, (A) and (B), they shall have a given ratio."

The writer of this notice has recently considered the same subject, in connection with circles of similitude, and our united labours form the subject of a paper printed in the appendix to the Lady's and Gentleman's Diary for 1855.

The fifteenth and sixteenth volumes are devoted to the consideration of the Mascheronian geometry, or that which is limited to the use of the ruler or the compass alone. He commences with the division and subdivision of lines, the division of arcs of circles, drawing tangents, and finding proportionals. He then proceeds to the description of polygons; their inscription in circles, and in each other-to many of which problems four or five different constructions are given. however, by far the most curious and valuable. cribing a tangential circle to touch two or more having given various constructions to these he proceeds to the construction

The latter volume is, He commences by desgiven circles, and after

of the different cases of the Apollonian problem of tangencies, with the exception of that where a tangent circle to three given circles is required. to be described; the enunciation of the problem being all that appears in the manuscript. The remainder of the volume is occupied with the construction of numerous other problems relating to the intersection of circles or tangents to them drawn from given points and having given ratios, many of which are remarkably curious and interesting. His objects throughout appear to have been to extend and diversify Mascheroni's methods, and in these respects he has succeeded to a greater extent than it is possible for any verbal statement to describe.

Volume xvii. is a short paper fully written out for the printer. It contains four different constructions and demonstrations to the problem of having "a point P, and two parallel lines AQ, BR, given in position, to determine the position of a line PQR, of section, making the rectangle, sum of squares, or difference of squares, of the segments AQ, BR, cut off from the lines given in position, equal to a given square; and was intended for insertion in the third number of the Apollonius.

The eighteenth volume is divided into two parts, the first of which is devoted to the solution of diophantine and other algebraical enquiries; and the second to the consideration of numerous original and selected geometrical problems. Pages 298-308 contain a discussion of the problem "to determine a point P in AC, the side of a given triangle ACB, such that drawing PQ perpendicular, and PR parallel to the base AB, the ratio, sum, difference, rectangle, sum of squares, or difference of squares, of PQ and PR, may be respectively equal to given quantities;" four different constructions and demonstrations being given to each case. The problem partially considered in volume xvii. is extended to the cases of the ratio, sum, or difference of AQ and BR, in pages 308-316; and several other problems are treated in a similar manner, each successive variation unfolding new properties of the illustrative diagrams, and affording additional proofs of Mr. Swale's extensive powers in geometrical research. A case of Apollonius on inclinations closes this volume, which, from internal evidence, appears to contain the latest efforts of his untiring mind.

In consequence of the absence of so many of the requisite demonstrations, an immense mass of Mr. Swale's speculations must ever remain in an incomplete and unprofitable condition. The state of geometrical

science has undergone a radical change since the time he wrote, and hence, few will hereafter be found either able or willing to supply such demonstrations as will render the theorems and problems intelligible to the student. It is also much to be regretted that he spent so much time on merely isolated problems, which have, at best, but little beyond their difficulty to recommend themselves to notice. As we have elsewhere observed, "his systematic researches on tangencies, maxima and minima, the inscription of polygons in circles and in each other, printed in his Apollonius, afford convincing proofs of how much he was capable of, when his powers were directed to regular subjects of enquiry; for the elegant methods of research employed in these papers, and the simplicity and beauty of the results obtained must ever command the admiration of geometers. His fertility of invention, and originality of conception, were inferior to those of no contemporary geometer, and had he directed those energies to systematic enquiries which he expended on the solution of some thousands of isolated and comparatively uninteresting questions, he might have systematized scattered topics or originated new theories in which he would have rivalled Carnot in transversals, Davies in spherics and porisms, or Chasles in anharmonic ratio, and have thus secured for his own name a permanent place in the history of modern geometry.

What will ultimately become of these manuscripts is, of course, beyond conjecture. That they will be almost religiously preserved by his son during his life no one will doubt who is acquainted with the profound veneration he entertains for the memory of his father;-but when we call to mind that a second generation has deliberately burnt the papers left by the Stewarts, and that already much of Mr. Swale's correspondence has been destroyed by an accidental fire, it may not be improper to suggest that these manuscripts ought to be deposited in some public library, where they would at once be safe and accessible, and like Dr. Simson's Adversaria, at Glasgow, ever remain an enduring monument of the genius and industry of so devoted a geometer."

REMARKS UPON THE FLORA OF LIVERPOOL.

By H. S. Fisher.

(READ 24TH MAY, 1855.)

The Flora of this district is continually presenting new aspects. Dock extensions and vast building operations destroy old and well known localities for plants; while our railways forming fresh sites, and disinterring buried seeds, add new varieties to our Flora. Another source of supply, equally fruitful, may be from the introduction of foreign or other seeds from a distance with our merchandise. Of these facts we have many instances. Some of our local botanists can recollect the time when specimens could be gathered on the site of some of the docks, now entirely surrounded with densely populous streets; while within the space of a few years, plants might have been gathered on the shore where now is formed our many miles of northern docks. Of those plants so lost we may mention the Convolvulus Soldanella, the sea-side convolvulus, formerly found abundantly near the Bootle land marks, but now entirely eradicated; and the still more rare Asparagus officinalis, the common asparagus, found in the same place, but now also totally lost. The plants of this district have received many interesting additions since the publication of our Liverpool Flora; of those plants I shall now give a list, with a few observations on the most interesting of them.

Helianthemum guttatum. This plant, although common in many parts of Lancashire, was not found in this locality until last year, when a small patch of it was found by Mr. Thomas Williams, on the sand hills below Halsall.

Viola lutea. The yellow violet or pansy, I am told, many years ago used to be found in this neighbourhood; and it is stated by Mr. Grazebrook in his Guide to Southport, to occur in that locality, but there are no specimens to substantiate his statement. The year before last I had it brought to me from Allerton, where, in a cultivated field it was growing in great abundance and luxuriance. The flowers vary from all yellow to dark purple and yellow.

* Dr. Dickinson gives it as his opinion that even in this new locality the plants have escaped from some garden.

Viola odorata, variety alba. The white sweet-scented violet grows abundantly in the grounds of a gentleman's house at Aintree. This is the only genuine wild locality we have for this general favourite; for although it is found in several situations, yet it is I believe, without exception planted.

Silene hirsuta. A hairy variety of S. inflata. This very pretty plant was found on the roadside between Bebbington and Parkgate. Some of the specimens were very luxuriant, two or three feet in height. I am not aware that this is the general appearance of the plant, if so, it would seem to point it out as a separate species, as this, combined with the hirsute margin of the leaves, gives it a very distinct appearance.

Stellaria glauca. This is in Mr. Aughton's list of Southport plants, but the locality was considered doubtful. Mr. Thomas Williams found it

plentiful in ditches on Martin Mere.

Cerastium atrovirens, of Babington's manual, a variety of Cerastium tetrandrum, though the difference is very trifling, and seems principally to consist in the bract of atrovirens having a very narrow membranous margin, while that of tetrandrum is broadly membranous. It is found on New Brighton sand hills. J. Shillitoe.

Erodium maritimum. Sparingly on a sandy hillock between Birkdale and Southport. T. Williams.

Polygala oxyptera is worthy of notice, for although only ranking as a variety, yet from its mode of growth and general appearance, it is easily distinguished from Polygala vulgaris, from which it differs in having smaller flowers, and the fruit broader than the wings of the calyx. The flowers vary in colour from pure white to a deep blue. On the sand hills at Waterloo it is abundant, but I have not observed it in any other locality along the coast.

Lathyrus Aphaca, the yellow vetchling. Of this rare species I found one specimen in 1852, on Seaforth Common near the Rimrose Bridge, but although I have searched diligently for it each season since, I have been unable to find it again. As this is a likely locality for it, it may in the course of time be re-discovered.

The want of a thorough examination of the plants of this neighbourhood I think, is fully proved by the fact that among the Rubi alone, I was enabled last season to add to our Flora no fewer than nine species. It is true, that