wonderful properties were more and more discovered, and the art continually gained ground and improved, by the discoveries of succeeding mathematicians.

This appears to be the most probable origin of this science; though Josephus seems to ascribe the invention to the Hebrews; while others of the ancients make Mercury the inventor. Polyd. Virg. de Inv. Rer. l. 1, c. 18.

Thales is said to have introduced this science from Egypt into Greece; where it was greatly cultivated and improved by himself, as well as by Pythagoras, Anaxagoras of Clazomenæ, Hippocrates of Chios, and Plato; who testified his conviction of the necessity and importance of geometry to the successful study of philosophy, by inscribing over the door of his academy, Let no one ignorant of geometry enter here. Plato thought the word geometry too mean a name for this science; and substituted instead of it the more extensive name of mensuration; and after him others gave it the title of pantometry. But even these are now become too confined in their import fully to comprehend its extent : for it not only enquires into, and demonstrates, the quantities of magnitudes, but also their qualities, as the species, figures, ratios, positions, transformations, descriptions, divisions, the finding of their centres, diameters, tangents, asymptotes, curvatures, &c.

About fifty years after Plato, Euclid collected together all these theorems, which had been invented by his predecessors in Egypt and Greece, and digested them into fifteen books, entitled the Elements of Geometry: demonstrating and arranging the whole in a very accurate and perfect

manner.

The next to Euclid, of those ancient authors whose works are extant, is Apollonius Pergæus, who flourished in the reign of Ptolemy Euergetes, about A. A. C. 230, and 100 years after Euclid. He was author of the first and principal work on Conic Sections; on account of which, and his other accurate and ingenious geometrical works, he acquired from his patron the emphatic appellation of the Great Geometri

cian.

Contemporary with Apollonius, or perhaps a few years before him, flourished Archimedes, celebrated for his extraordinary mechanical inventions during the siege of Syracuse, and no less so for his many ingenious geometrical_compositions. Eudoxus of Cnidus, Archytas of Tarentum, Philolaus, Eratosthenes, Aristarchus of Samos, Dinostratus the inventor of the quadratrix, Menechmus his brother and the disciple of Plato, the two Aristæuses, Conon, Thracidius, Nicoteles, Leon, Theudius, Hermotimus, Hero, and Nicomedes, the inventor of the conchoid, besides many other ancient geometricians, have contributed to the improvement of geometry.

The Greeks continued their attention to it, even after they were subdued by the Romans; whereas the Romans themselves were so little acquainted with it, even in the most flourishing time of their republic, that Tacitus informs us they gave the name of mathematicians to those who pursued the chimeras of divination and judicial astrology. Nor does it appear they were

disposed to cultivate geometry during the decline, and after the fall of the Roman empire. But the case was different with the Greeks; among whom are found many excellent geometricians since the commencement of the Christian era, and after the translation of the Roman empire. Ptolemy lived under Marcus Aurelius; and we have still extant the works of Pappus of Alexandria, who lived in the time of Theodosius; the commentary of Eutocius, the Ascalonite, who lived about A. D. 540, on Archimedes's mensuration of the circle; and the commentary on Euclid by Proclus, who lived under the empire of Anastasius.

The subsequent inundation of ignorance and barbarism was unfavorable to geometry, as well as to the other sciences; and the few who applied themselves to this science were calumniated as magicians. But, in those times of European darkness, the Arabians were distinguished as the guardians and promoters of science; and from the ninth to the fourteenth century they produced many astronomers, geometricians, geographers, &c.; from whom the mathematical sciences were again received into Spain, Italy and the rest of Europe, somewhat before the year 1400.

Some of the earliest writers, after this period, are Leonardus Pisanus, Lucas Pacciolus or de Burgo, and others between 1400 and 1500. And after this appeared many editions of Euclid, or commentaries upon him: thus Orontius Finæus, in 1520, published a commentary on the first six books; as did James Peletarius in 1556; and about the same time Nicholas Tartaglia published a commentary on the whole fifteen books. There have been also various other editions, or commentaries; but the completest edition of all the works of Euclid is that of Dr. Gregory, printed at Oxford in 1703, in Greek and Latin. The edition of Euclid by Dr. Robert Simpson, of Glasgow, containing the first six books, with the eleventh and twelfth, is much esteemed for its correctness; and Playfair's edition of the first six books, with two additional ones on solids, the intersection of planes, and the quadrature of the circle, is remarkable for the precision and taste which distinguished every thing that came from the pen of that accomplished philosopher.

Besides the different editions of Euclid, we have several other elementary treatises on geometry; the principal of which are those by Emerson, Simpson, Legendre and Leslie. All these are works of considerable merit and usefulness. Emerson's is quite a store-house of properties; Simpson's is distinguished by an elegant tract on geometrical maxima and minima, and by a copious collection of geometrical problems, with their solutions, exceedingly well adapted to improve the dexterity and to form the taste of the young student. Leslie's work contains the best introduction to geometrical analysis that is to be met with in the English language. Legendre's work, which has been recently translated into English, is, in many respects, exceedingly valuable; but there appears little reason to expect that it will supplant, in the English schools, the elements of Euclid, which, after all that has been done on the subject, is, for the purposes of tuition, still unrivalled.

Among those who have gone beyond Euclid, in the application of the elementary principles of geometry, may be named Apollonius, in his Conics, Plane Loci, Determinate Section, Tangencies, &c.; Archimedes, in his Treatises on the Sphere and Cylinder, the Dimensions of the Circle, of Conoids, Spheroids, Spirals, the Quadrature of the Parabola, &c.

Since the application of algebra to the geometry of curve-lines, this science has been greatly extended, and we may number, among those who have contributed to its improvement, almost every name of eminence connected with abstract science. The, names of Descartes, Schooten, Leibnitz, Bernouilli, Maclaurin, Cotes, and Waring, are intimately associated with the history of this branch of the science. The most valuable work, however, on the algebraic geometry, is one recently published by Dr. Lardner, of Dublin.

On the subject of practical geometry, the writers may truly be said to be numberless; the chief are Beyer, Kepler, Ramus, Clavius, Mallet, Tacquet, Ozanam, Gregory, and Hulton.

On the whole, the history of geometry_may be divided into four grand eras, viz. 1st, From its invention to its introduction into Greece by Thales; 2d, From that period to the time of Euclid; 3d, From Euclid and Archimedes to the application of algebra to the subject by Descartes; and 4th, From Descartes to Newton and to the present time, a period in which the invention of fluxions, and their application to this and other branches of science, has cast all previous discoveries into shade.

This science is usually divided into two parts, theoretical geometry, and practical geometry, the former showing the principles of the science, and the latter their application. For their application, however, to MENSURATION, TRIGONOMETRY, and NAVIGATION, we shall refer to the articles under those titles; and give in this article their application in the solution of practical problems in pure geometry.

PART I.

THEORETICAL GEOMETRY; OR GENERAL PRINCIPLES OF THE SCIENCE,

DEFINITIONS.

1. A POINT is that which has position but not magnitude.

2. A LINE is length without breadth or thickness; the extremities of a line are therefore points.

3. A RIGHT LINE, or STRAIGHT LINE, is that which lies evenly between its extreme points. Fig. 1, Plate X.

4. A SUPERFICIES is that which has only length and breadth; the extremities of a superficies are therefore lines, and the intersections of superficies with one another are also lines.

5. A PLANE SUPERFICIES is that in which any two points being taken, the straight line between them lies wholly in that superficies.

6. A PLANE RECTILINEAL ANGLE is the inclination of two straight lines to one another, which meet together, but are not in the same straight line. Fig.2.

Note. When several angles are formed at the same point, as at B fig. 3, each particular angle is described by three letters, whereof the middle one shows the angular point, and the other two the lines that form the angle, thus CBD or DBC denotes the angle contained by the line C B and DB.

7. When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a RIGHT ANGLE, and the straight line which stands on the other is called a PERPENDICULAR. Fig. 4.

8. An OBTUSE ANGLE is that which is greater than a right angle. Fig. 5.

9. An ACUTE ANGLE is that which is less than a right angle. Fig. 6.

10. PARALLEL STRAIGHT LINES are such as are in the same plane, and which being produced ever so far both ways do not meet. Fig. 7.

11. A FIGURE is that which is enclosed by one or more boundaries.

12. RECTILINEAL FIGURES are those which are contained by straight lines.

13. Every plane figure bounded by three straight lines is called a TRIANGLE, of which the three straight lines are called the sides, that side upon which the triangle is conceived to stand is called the base, and the opposite angular point the vertex.

14. An EQUILATERAL TRIANGLE is that which has three equal sides.__ Fig. 8.

15. An ISOSCELES TRIANGLE is that which has only two equal sides. Fig. 9

16. A SCALENE TRIANGLE is that which has all its sides unequal. Fig. 11.

17. A RIGHT-ANGLED TRIANGLE is that which has a right angle. Fig. 10.

18. An OBTUSE-ANGLED TRIANGLE IS that which has an obtuse angle. Fig. 11.

19. An ACUTE-ANGLED TRIANGLE is that which has all its angles acute. Fig. 12.

20. Every plane figure bounded by four straight lines is called a QUADRILATERAL, and the right line joining the opposite angles is called a DIAGONAL.

21. A PARALLELOGRAM is a quadrilateral of which the opposite sides are parallel. Fig. 13. 22. A RECTANGLE is a parallelogram which has one right angle. Fig. 14.

23. A SQUARE is a rectangle which has all its sides equal. Fig. 15.

24. A RHOMBUs is a parallelogram which has all its sides equal. Fig. 17.

25. A TRAPEZIUM is a quadrilateral which has not its opposite sides parallel. Fig. 18.

26. A TRAPEZOID is a quadrilateral which has two of its opposite sides parallel Fig. 19.

27. Plane figures bounded by more than four straight lines are called POLYGONS. Fig. 16.

28. A PENTAGON is a polygon of five sides, a HEXAGON has six sides, a HEPTAGON seven, an OCTAG ON eight, a NONAGON nine, a DECAGON ten, an UNDECAGON eleven, and a DODECAGON twelve sides.

29. A REGULAR POLYGON has all its sides and all its angles equal; if they are not equal, the polygon is IRREGULAR.

30. A CIRCLE is a plane figure bounded by one

line called the circumference, which is such, that all straight lines drawn to it from a certain point within it called the centre are equal; and these straight lines are called the radii of the circle. The circumference itself is also often called a circle. Fig. 20.

31. The DIAMETER of a circle is a straight line passing through the centre, and terminated both ways by the circumference.

32. An ARC of a circle is any part of its circumference. Fig. 21.

33. A CHORD is a straight line joining the extremities of an arc. Fig. 21.

34. A SEGMENT is any part of a circle bounded by an arc and its chord. Fig. 21.

35. A SEMICIRCLE is half the circle, or a segment cut off by a diameter. The half circumference is also sometimes called a semicircle. Fig. 20.

36. A SECTOR is any part of a circle which is bounded by an arc and two radii drawn to its circumference. Fig. 22.

37. A QUADRANT, or quarter of a circle, is a sector having a quarter of a circle for its arc, and its two radii perpendicular to each other. A quarter of the circumference is also called a quadrant. Fig. 23.

38. The HEIGHT, or ALTITUDE, of a figure is a perpendicular let fall from an angle or its vertex to the opposite side or base. Fig. 24.

39. In a right-angled triangle the side opposite the right angle is called the HYPOTHENUSE, and the other two sides are called the LEGS, or sometimes the base and perpendicular. Fig. 10. 40. The circumference of every circle is supposed to be divided into 360 equal parts, called DEGREES, and each degree into sixty MINUTES, each minute into sixty SECONDS, and so on. Hence a semicircle contains 180 degrees, and a quadrant ninety degrees.

41. The MEASURE OF A RECTILINEAL ANGLE is an arc of any circle contained between the two lines which form that angle, the angular point being the centre, and it is estimated by the number of degrees in that arc. Fig. 25.

42. IDENTICAL FIGURES are such as have all the sides and all the angles of the one, respectively equal to all the sides and all the angles of the other, each to each, so that if the one figure were applied to, or laid upon the other, all the sides of the one would exactly fall upon and cover all the sides of the other, the two becoming, as it were, but one and the same figure.

47. A TANGENT TO A CIRCLE is a straight line that meets the circle at one point, and every where else falls without it. Fig. 27.

48. A SECANT is a straight line that cuts the circle; lying partly within and partly without it. Fig. 27.

49. A RIGHT-LINED FIGURE is inscribed in a circle; or the circle circumscribes it, when all the angular points of the figure are in the circumference of the circle. Fig. 28.

50. A RIGHT-LINED FIGURE circumscribes a circle; or the circle is inscribed in it, when all the sides of the figure touch the circumference of the circle. Fig. 28.

51. ONE RIGHT-LINED FIGURE is inscribed in another; or the latter circumscribes the former, when all the angular points of the former are placed in the sides of the latter. Fig. 28.

52. SIMILAR FIGURES are those that have all the angles of the one equal to all the angles of the other, each to each, and the sides about these angles proportional.

53. The PERIMETER OF A FIGURE is the sum of all its sides taken together.

Note. When the word line occurs, without the addition of either straight or curved, a straight line is always meant.

AXIOMS.

1. Things which are equal to the same thing are equal to one another.

2. When equals are added to equals, the wholes are equal.

3. When equals are taken from equals, the remainders are equal.

4. When equals are added to unequals, the wholes are unequal.

5. When equals are taken from unequals, the remainders are unequal.

6. Things which are like multiples of the same thing are equal to one another.

7. Things which are like parts of the same thing are equal.

8. The whole is equal to all its parts taken together.

9. Things which coincide, or fill the same space, are identical, or mutually equal in their parts.

10. All right angles are equal to one another. 11. A line which meets one of two parallel lines will if produced meet the other.

12. If two straight lines intersect each other, they cannot both be parallel to the same straight

PROPERTIES OF STRAIGHT LINES, and Plane

RECTILINEAL FIGURES.

THEOREM I.-In any two triangles, as ABC, DEF (fig. 30 p. I), if two sides, as A B, AC, in the one, be respectively equal to two sides, as DE, DF, in the other, and the angle A, included by the sides A B, AC, be equal to the angle D, included by the sides DE, DF; then the triangles are equal in all respects, and nave the angles equal which are opposite the equal sides.

For, conceive the point A to be laid on the point D, and the line A B on the line DE, then, as these lines are equal, the point B will fall on the point E. And as A B coincides with DE