5. Several of the principles on which the Levelling and Surveying of Land, and Geographical and Astronomical Observations depend, have been given in the Third Book,-such as the Methods:

1. Of computing the distances and heights of objects when situated on the verge of the natural horizon; Use 16, III.

2. Of determining the part of a globe which may be enlightened by a luminous body, as by a meteor, volcano, lighthouse, &c. ; of explaining the theory of the phases of the moon; of ascertaining the distance of the sun; and of obtaining the dip of the horizon ; Use 19, III.

3. Of constructing a figure representative of the distance of the place of observation from an object; and of tracing the arc of a circle for giving a spherical figure to optical glasses; Use 21, III.

4. Of finding the true centre of an imperfectly constructed theodolite,
or similar instrument; Use 26, III.

5. In Coast Surveying, for noting soundings, bearings, &c.; Use 33, III.
6. For taking a true level on the earth's surface; Use 36, III.; and
7. For calculating the earth's diameter; Use 37, III.

These are but Examples of the many useful purposes to which geometrical science may be applied; and they may serve to redeem geometry from the prejudiced objection that it is a system of theoretical reasoning without practical results. The practical results are really most important, and in the actual business and occupations of life are an everyday's demand.

GRADATIONS IN EUCLID.

BOOK IV.

STRAIGHT

CONTAINING THE METHODS OF CONSTRUCTING REGULAR
LINED FIGURES IN AND ABOUT A CIRCLE, AND CIRCLES IN
AND ABOUT REGULAR STRAIGHT-LINED FIGURES.

Excepting Prop. A, Theor, the fourth book of Euclid's Plane Geometry consists entirely of Problems; it is in fact the Application of the third book to the purposes of inscribing and circumscribing triangles and other regular straight-lined figures in and about circles, or circles in and about such regular figures. The former books supplied the means of drawing regular plane figures of 3, 4, 5, and 15 sides; and by continued bisections of making them of 6, 12, 24, &c., or 8, 16, 32, &c.; or 10, 20, 30, &c., sides. The employment of those means constitutes the object of the book on which we are now entering.

In Trigonometry, Astronomy, and the various departments of Civil and Military Engineering, the fourth book is found of essential service; we also deduce from it the method of obtaining, with sufficient exactness, the quadrature of the circle, and of proving that circles are to one another in the proportion of the squares of their diameters.

DEFINITIONS.

1. A rectilineal figure is said to be inscribed in another rectilineal figure when the angular points of the inscribed figure touch the sides of the figure in which it is inscribed, each upon each.

E

H

B

Thus, the fig. ABCD is inscribed in the fig. EFGH.

2. In like manner, a figure is said to be F described about another figure when the sides of

the circumscribed figure touch the angular points of the figure about which it is described, each upon each;

Thus the fig. EFGH is circumscribed about the fig. ABCD.

It is noteworthy that EUCLID gives no example of one rectilineal figure being inscribed in another rectilineal figure, or circumscribed about it.

3. A rectilineal figure is said to be inscribed in a circle when each angular point of the inscribed figure touches the circumference of the circle;

Thus, the qu. lat. A CBD is inscribed in the circle ADBC.

4. A rectilineal figure is said to be described about a circle when each side of the circumscribed figure touches the circumference of the circle;

Thus, the qu. lat. EFGH is described about the
circle ABCD.

5. In like manner, a circle is said to be inscribed in a rectilineal figure when the circumference of the circle touches each side of the figure;

Thus, the circle ABCD is inscribed in the quadrilateral EFGH.

6. A circle is said to be described about a rectilineal figure when the circumference of a circle touches each corner of the figure about which it is described;

Thus, the circle ABCD is described about the figure ADBC.

7. A straight line is said to be fitted exactly into a circle, or to be applied in it, when the extremities of it are on the circumference of the circle;

Thus, the lines AC and AD are applied to the circle ABCD.

DEFINITIONS ADDITIONAL TO THOSE OF EUCLID.

8. A circle is said to be exscribed to a triangle when, having for centre the point of intersection of any two straight lines that bisect the exterior angles of the triangle, the circle touches a side of that triangle.

9. "Any rectilineal figure, of five sides and angles, is called a pentagon; of six sides and angles, a hexagon; of seven sides and angles, a heptagon; of eight sides and angles, an octagon; of nine sides and angles, a nonagon, of ten sides and angles, a decagon; of twelve sides and angles, a duodecagon; of fifteen sides and angles, a quindecagon," &c.

10. "These figures are included under the general name of polygons; and are called equilateral when their sides are equal; and equiangular when their angles are equal. Also, when both their sides and angles are equal they are called regular polygons." POTTS' EUCLID, p. 124.

N.B.-The force of the propositions in Simpson's Edition is often lessened by not rendering the Greek original into English corresponding, as far as differences of idiom will admit, more closely with Euclid's text. To avoid this, Galbraith and Haughton's rendering of the general enunciation is often followed, though they have not been so thoroughly exact as is desirable.