were suffered to remain in total oblivion. From this state of degradation they were extricated by the Arabians, who transplanted and naturalized them where they obtained nourishment and honour. Thus it came to pass, by a singular revolution, that the destroyers of knowledge were converted into its preservers. The Italians, by their intercourse with Arabia, became, about the end of the twelfth century, the restorers of its long-forgotten Arts and Sciences to Christendom. Leonardus Pisanus, a rich merchant of Pisa, who had made in the course of his profession several voyages to the East, is said to have been the first who disseminated amongst Europeans a knowledge and a love of mathematical studies after both had disappeared for so many ages*.

Since the revival of learning in Europe, Geometry has obtained a due portion of attention, and has indeed ever been the favourite study of men eminent for scientifical genius. Not a few have emulated with considerable success the most illustrious of the ancients; none, however, have excelled them, and but one has equalled themNEWTON. He is the "Great Geometer" of the new ages, and may be styled the modern Archimedes, as Archimedes is frequently designated the Newton of antiquity. But the chief improvements made in this Science by the moderns regarded its higher branches; they were rather accessions than alterations, and tended rather to promote than perfect it. As much as could be done in the latter way was considered to have been already performed by Euclid in his "Elements," and Apollonius in his several treatises. Of the former, in which only our readers are particularly interested, a short account may be acceptable.

Alexandria and Tyre dispute for the honour of having given birth to Euclid, the mathematician, who however is ascertained to have flourished and taught at the former place about the year 280 B. C. Besides other works of less celebrity, he compiled the "Elements" which go by his name. Opinions are different as to what share he had in this system of Geometry, but it is allowed on all hands

* A manuscript, dated 1202, of this celebrated Italian has lately been discovered by Cossali, a canon of Parma.

that to him is due the peculiar disposition and arrangement of its materials. He probably reduced to their present order in that work the several principles discovered or introduced by Thales, Pythagoras, Plato, and other philosophers before him, adding and inserting many of his own. The Elements are divided into fifteen books, of which, however, the last two are suspected to have been annexed by a different person two centuries later. All but four of these (the 5th, 7th, 8th, and 9th) are on the subject of Geometry alone; the principles of Numbers are delivered in the 5th, 7th, and 8th; while the 5th regards both these sciences. None but the first six books, which treat of geometrical magnitudes in the same plane, with the 11th and 12th, which treat of Solids, are now studied in Universities; the use of the others being superseded by modern improvements in Arithmetic.

Euclid's Elements have long remained the text-book of Geometry; and it is only within these few years that their perfection has been doubted, and the possibility of their improvement suggested. Many systems of elementary Geometry, distinguished by acuteness and elegance, but none by superiority over that of Euclid, are now extant. If it were only from the circumstance of the authority of the latter as a text-book having been so long recognised, it will probably be recognised as such still longer.

In the above concise sketch of the origin and progress of Geometry we have deviated as little as possible from the straight line of history. Whatever collateral facts or anecdotes remain shall be interspersed by detail through our little Volume, so as to render it as attractive as is compatible with a subject not easily rendered at once useful and entertaining.

After what has been said, it is we hope needless to implore the reader that he will extirpate from his mind the notion that book Geometry is an arbitrary creation of certain speculative philosophers, which has no connexion with any thing in existence, and to understand which it is requisite that we should lay aside all we have learned from our senses, and begin a new life as it were of pure intellectuality. Geometry is indeed an elevated structure, but it has its foundation on the Earth. The gross air about us refines itself by degrees into a still clearer and clearer at

mosphere, till in the uppermost regions it becomes pure ether; this, however, for all its altitude and transparency, rests upon the succession of lower and denser atmospheres, and is ultimately supported by the lowest and densest of all. So likewise with Geometry: however refined and exalted at length, it is at first as rude and humble as our commonest notions, which are the basement upon which it is gradually raised to such a height and magnificence.

It is to be observed, however, that many of our Elements being merely introduced as stepping-stones to others, their practical uses and applications are to be considered as involved and mixed up with those of the latter, affording none such by themselves: that many, likewise, have practical uses and applications which are too refined and remote either for us to communicate or our readers to understand without descriptions and prefatory matter totally inconsistent with the nature of our little treatise: and that, finally, we neither pretend nor propose to exhibit all the uses and applications, even of the simple kind, derivable from our elementary principles, but merely such as will accomplish the design set forth in the Prospectus of our COMPANION LIBRARY, namely—"To confirm its readers in the Elements of our first Series, to instruct them in the method of drawing practical benefit from their acquired knowledge, and to furnish a ready answer to the often repeated inquiry-What is the use of abstract Science?"

We shall follow the division into PARTS and LESSONS adopted in our Popular Geometry.

PART I.

B

DEFINITION I. "A rectilineal (or right-lined) angle is that which is formed by two right lines meeting at the same point, but not lying in the same right line." The instrument called a Bevel, of extensive use in the mechanical arts, is a practical instance of an angle. It is formed of two straight blades OE, OD, turning on the pivot o, and opening to any width, DXE, which may be desired. This

E

instrument is of great use in delineating, or measuring, or transferring angles. For example, if we wished to delineate the angle which the mast

of a ship made with the deck, we have only to open the blades, so as that one, OD, shall lie along the mast, the other, OE, along the deck, when the pivot o is thrust

between the mast and deck. Then taking up the instrument, without moving either blade, lay it on a flat surface and trace its outer edges: this will evidently give the inclination of the mast, i. e. in the direction where the bevel stood. If that direction were exactly "fore and aft" (as the sailors term it), or exactly along the middle line of the deck from stem to stern, the angle delineated would be what is called the rake of the mast.

In the same manner an angle is measured, or transferred, -the blades of the instrument moving stiffly round the pivot, so as when once set to an angle not to change it without some compulsion. A pair of compasses, if the legs were pretty straight, and the hinge pretty stiff, might serve upon occasion instead of a bevel, though we cannot recommend what might be so injurious to the substitute.

What Anatomists call the "facial angle" is the angle formed by the slope of the fore

head with the direction of the upper row of back teeth*. Thus, if BA be a straight line from the most prominent part of the forehead to A, the most prominent part of the upper front teeth, and if AC be another straight line from the point a in the direction of the upper back teeth, then the line BA is the facial line, and the angle BAC the facial angle, (from facies,

* We use this description as more generally intelligible than the professional one, which it would be hard to render clear without technical

terms.

the face). In the above profile, which is that of Cicero, the facial line is nearly perpendicular to the horizontal line AC.

Now, the more prominent the forehead with respect to the upper jaw, the greater the facial angle, as is evident; and hence the size of the facial angle becomes a test of the longitudinal capacity of the forehead, and thence in some measure of the intellectual powers which are supposed to reside in, and be commensurate with, the forepart of the skull. Of all the animals, the most rational, Man, has the greatest facial angle; in him it frequently amounts to 90o, or a right angle, in which BA is perpendicular to AC. It is sometimes, however, so low in the human subject as 65o, which is the facial angle of the Ourang-outang, thus linking us to the species next in intelligence beneath us. As we descend in the scale of rational beings the facial line becomes more and more depressed; the front teeth become more protruded, and the forehead leans farther back, as may be seen in dogs, horses, birds, &c.; till in reptiles and fishes the line BA has scarcely any inclination to AC, and therefore the facial angle vanishes nearly altogether,-betokening the lowest degree of stupidity.

Observe an antique statue of a celestial deity: its facial angle is often 100o, as appears in the opposite bust, which is that of Jupiter. By this the Greeks, who had gathered from experience and observation that a great development of forehead betokened expansive powers of judgment and imagination, proposed to indicate the superhuman faculties of their Gods. To the fanciful busts of their heroes they usually gave a facial angle of 90°, as may be seen in

that of Theseus among the Elgin Marbles. And to those of all their great men, who approached the gods in genius or wisdom, (Pythagoras, for example, whose profile is