GEOMETRY. That branch of mathe matics which treats of the properties of ex tension and figure Geomeerris distinguish ed into thetheoretical and the practicaL Theo retical or speculative geometry treats of the various properties and relations in magnitudes, etc. Practical geometry comprehends the construction of figures, the drawing of tines in certain positions, as parallel or perpen dicular to each other, dm. Speculative geome try is again distinguished into elementary geometry, that treats of the properties and proportions of right lines and right lined figures, as also of the circle and its several parts ; and the sublime or transcendental geometry, that treats of the higher order of curves, Ste The simple principles of geometry are ex plained in definitions and axioms. The fol lowing are the most important definitions : A point is that which has neither length, breadth, nor thickness - a line has length without breadth or thicinm ; a superficies, or surface, has length and breadth only, the boundaries of which am lines ; a solid has length, breadth, and thickness; the boundaries of a solid are surfaces. A straight line lies evenly between the parts, parallel lines keep at the same dis tance from each other when extended indefi nitely. A perpendicular line is perpendicular to another line. An angle is formed by the meeting of two lines in a point ; it is a right angle when farmed by one line falling perpen on another line; an obtuse angle, when it as greater than a right angle; and an acute angle when it is less. A figure is a space included within one or more boundaries, called sides ; it is rectilinear when contained by right lines, and curvilinear when contained by curved lines ; a rectilinear figure contained by three right lines is a triangle ; if by Raw, quadrilateral; if by five, a pentagon ; if by six, a hexagon, etc. ; if by more than twelve, a polveus Triangles are distinguished according to the length of their sides, into equilateral, having all the sides equal ; isosceles, two sides equal ; and scalene, having all the sides unequal; or, according to their angles, into right angled, if they have one right an gle; obtuse angled, if they have one obtuse angle; . and acute anOetl, if have all acute angles. Every or four-sided figure is called a parallelogram when it has its sides parallel, and a rectangle when all its angles are right angles. Four-sided figures are, moreover, distinguished according to their sides and angles, into a square, which has all its sides equal and its angles right ones; an oblong square, which has its oppouite sides equal and its angles right ones; a rhombus, having all the sides equal, but the angles not right ones ; and a rhomboid, having the op posts sides equal, and the angles not right ones. When a quadrilateral has none of its sides parallel it is a trapezium, and when only two of its skies parallel a trapezoid. The dia
gonal is the right line which divides a paral lelogram into two equal parts. The base of a figure is the side on which it is supposed to stand. The vertex is the extreme point op posite to the base; the altitude is the perpen dicular distance from the vertex to the Lase. The area of a figure is the quantity of space contained within its boundaries.
Of curvilinear figures in common geometry is the circle, which is a plane figure bounded by a curve line called the circumference, which is equally distant from a point called the cen tre. The diameter of a circle is a straight line drawn from one side of the circumference to the other, through the centre, so as to divide it into two equal parts. The radius of a circle is a straight line drawn from the centre to the circumference: the segment of a circle is a part cut off by a line called the chord. The circumference of every circle is supposed to be divided into 360 equal parts, called de grees, erery degree into 60 parts, called mi nutes, and every minute into 60 parts, called seconds.
Solids are distinguished into a prism, the sides of which are parallelograms, and the two ends or bases are similar ; polygons, parallel to each other; the cube, conszsting of six equal square sides or faces ; the pyramid, having any plane figure for its base and triangles for its sides, all terminating in one common point or vertex ; the cylinder, which is generated by the rotation of a rectangle about one of its sides supposed to be at rest ; the cone, a solid having a circular base, and its other extremity terminated in a single point or vertex. Those curves which are formed by the intersection of a plane with a cone form the subject of conic sections, which is a branch of sublime geometry.
Ratio is the mutual relation of two magni tudes of the mine kind to one another, in re spect to quantity, as 2 to 1, which is double ; the firmer of these is called the antecedent, and the latter the consequent : proportion is the similitude of ratios, as 6 is to 2 as 3 is to I, that is a triple ratio in both cases.
An axiom is a plain truth that wants no de monstration, as that the whole is greater than a part. A postulate is that which requires to be granted as true before another thing can be demonstrated. A proposition is that which proposes something to be done or demonstra ted; it is a problem when it proposes any thing to be to divide a given line into two equal parts, or to raise a perpendicular, die. ; and a theorem when it proposes some thing to be shown, as that triangles of the same base and altitude are equal to each other, or that all the angles in the same segment of an arch are equal, &o.