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allel to the base of the paper. In many good drawings, the horizon is not drawn parallel to the base, because positions are given to objects which would require more paper than could be spared to assign to the horizon its true position; but this is never forgotten either by the experienced draughtsman, or the judicious connoisseur.

4. A vertical line is that which presents the greatest contrast possible to a horizontal line, being always perpendicular to the true horizon; thus, if a weight be freely suspended from a fine hair, when the oscillation has ceased, the hair will present a near approximation to a vertical line, since the weight will, (by the attraction of gravitation,) invariably tend towards the centre of the earth.

5. An Oblique line, is any straight line which is neither vertical nor horizontal.

In pure geometry, any line is said to be oblique to another line, when the two contain an Oblique Angle, i. e., one which is either greater, or less than a right angle.

6. A Tangent is a line that touches a circle, or any other curve without cutting it; also, a line or circle is tangential, or is a tangent to a circle, or other curve, when it touches it, without cutting, although both be produced.

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The place where they touch is called the point of contact.

7. A Secant is a line which cuts a circle, lying partly within, and partly without it.

V. A Superficies is that which hath only length and breadth.

The diagrams to illustrate definitions XV. and XXIV. are superficies, but those of definitions IX. and X. are not.

VI. The extremities of a superficies are lines.

Superficies are either Plane, Concave, or Convex.

8. If a Superficies be bounded by right lines, it is recti− lineal; if by curved lines, curvilineal; and if by right and curved lines, mixtilineal.

VII. A Plane Superficies is that in which any two points being taken, the straight line between them. lies wholly in that superficies.

Every plane superficies is completely flat or even, and is either Horizontal, Vertical, or an Inclined plane.

9. A Concave superficies is that which is curvilineally



10. A Convex superficies is that which is globular.

Angles are either Plane or Spherical.

VIII. "A Plane Angle is the inclination of two lines to one another in a plane, which meet together, but 66 are not in the same direction."

Plane Angles are either Rectilineal, Curvilineal, or Mixtilineal.

IX. 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.


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N. B. When several angles are at one point B, any one of them is expressed by three letters, of which 'the letter that is at the vertex of the angle, that is, at 'the point in which the straight lines that contain the angle meet one another, is put between the other two 'letters, and one of these two is

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'somewhere upon one of those straight lines, and the other upon the other line: Thus, the angle B 'which is contained by the straight

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'lines AB, CB, is named the angle ABC, or CBA; 'that which is contained by AB, DB, is named the an

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gle ABD, or DBA; and that which is contained by DB, CB, is called the angle DBC, or CBD; but, if 'there be only one angle at a point, it may be expressed 'by a letter placed at that point; as the angle at E.'

11. A Curvilineal Angle is that whose legs are curved lines.

12. A Mixtilineal Angle is that which is contained by a right line and a curve.

13. A Spherical angle is that which is formed on the surface of a sphere, by the intersection of two of its great circles.

X. 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 to it.

Beginners are apt to suppose that a perpendicular, must be a vertical, or plumb line, which it need not be.

XI. An obtuse angle is that which is greater than a right angle.

XII. An acute angle is that which is less than a right angle.

The length of the legs of a rectilineal angle does not affect its quantity as an angle, for whether the legs be an inch or a mile in length, the angle remains unaltered, the measure of an angle, whose legs are right lines, being the arc which is contained between those lines, the vertex being its centre, and the radius taken at pleasure.

XIII. "A term or boundary is the extremity of any thing."

XIV. A figure is that which is enclosed by one or more boundaries.

It is customary as a matter of convenience, to refer to any diagram, as fig. 1, &c.; but geometrically, that only is a figure, which has either superficial or solid content.

XV. A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another.

XVI. And this point is called the centre of the circle.

A circle is the superficies or space enclosed by its circumference.

14. A great circle is that plane which divides a sphere into two equal parts; as the equator, ecliptic, or meridian, on a geographical globe.

15. A line drawn from the centre to the circumference of a circle, is called a Radius; and it may here be remarked further, relative to the content of an angle, that (as its measure entirely depends upon the properties of the circle,) it is customary with English mathematicians to divide the circumference of any circle, into 360 equal parts, which are named degrees; if therefore, one of these parts be contained by two radii, the angle formed by these radii will contain one degree; and in like manner, a right angle contains 90 degrees; because the arc which measures it is equal to one fourth part of the circumference of a circle. It will also be manifest that the lineal length of a degree depends entirely upon the length of the whole circumference; for example, the length of a degree upon the Earth (which is a fraction more than 69 miles), results simply from its circumference being nearly 25,000 miles in length; it may here also be noted, that, although all degrees of latitude on the artificial terrestrial globe are of equal length, yet those of longitude are not so, for only such as are on the equator, are equal in length to those of latitude; as it will be evident from an inspection of the globe, that, since the meridian lines intersect each other at the poles, the lineal length of each degree of longitude, depends upon its distance from the equator.

XVII. A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.

XVIII. A semicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter.

16. An Arc of a circle is any part of the circumference.

17. A Chord is a right line joining the extremities of an arc.

18. A Sector is any part of a circle which is bounded by an arc, and two radii drawn to its extremities.

19. A Segment is any part of a whole; thus, if a right line be cut into two or more parts, each part is a segment of the whole line.

XIX. "A segment of a circle is the figure contained "by a straight line, and the circumference it cuts off."

20. A quadrant is a quarter of a circle contained by two radii, which will of necessity be at right angles to one another.

Every semicircle is a segment, and every diameter of a circle is a chord; every quadrant is also a sector, and a sector may contain more or less, but can never contain exactly 180 degrees.

21. The complement of an arc or angle, is the difference between it and an arc or angle of 90 degrees; thus, the complement of an arc or angle of 30 degrees, is an arc or angle which contains 60 degrees; 20 degrees are the complement of 70 degrees, and so on,

22. The supplement of an arc or angle is the difference between it and 180 degrees; thus the supplement of 40 degrees, is an arc or angle of 140 degrees.

XX. Rectilineal figures are those which are contained by straight lines.

XXI. Trilateral figures, or triangles, by three straight


XXII. Quadrilateral, by four straight lines.

XXIII. Multilateral figures, or polygons, by more than four straight lines.

The sides of a trilateral figure must of necessity contain three angles, those of a quadrilateral, four; and of a multilateral, as many angles as there are sides to the polygon; when, therefore, the angles are the objects of investigation, the first figure is called a triangle, the second a quadrangle, and the third a multangle.

Triangles are either plane or spherical.

23. A plane triangle is that which coincides with a plane superficies.

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