The subscribers have examined, in manuscript, the additions to the seventh edition of FLINT'S SURVEYING, by GEORGE GILLET, Esq., Surveyor General of Connecticut, and find them to embrace a system of correct, useful, and practical matter, judiciously arranged, and clearly explained to the understanding of the learner. Having long acted as Surveyors under public authority, we recommend this work, as containing all the elementary science, and requisite tables, necessary or convenient for the learner and the practitioner. The present is a more full and complete system than any former edition.

MOSES WARREN, Dep. Sur. N. London Co.
LEMUEL INGALLS, late Dep. Sur. Windham Co.
DANIEL ST. JOHN, Dep. Sur. Hartford Co.
ASAHEL DEWEY, County Sur., N. London Co.
JONATHAN NICHOLS, Dep. Sur. Windham Co.

Connecticut, August, 1832.

FLINT'S SURVEYING has now been before the public upwards of thirty years. During this period, it has passed through numerous editions, and been enriched, from time to time, by important contributions from the present Surveyor General, GEORGE GILLET, Esq. The distinguishing feature of the work, as now published, is its excellent adaptation to the every day wants of the practical surveyor, while it supplies to academies and private students an eminently useful, clear, and well-digested system of elementary instruction, both in the theory and practice of Surveying. I know of no work, in this respect, which equals it. É. H. BURRITT, Civil Engineer.

New Britain, Con., Nov., 1835.

GEOMETRY.

GEOMETRY is a science which treats of the properties

of magnitude.

Ꮲ Ꭺ Ꭱ Ꭲ I .

GEOMETRICAL DEFINITIONS.

1. A point is a small dot; or, mathematically considered, is that which has no parts, being of itself indivisible. 2. A line has length but no breadth.

3. A superficies or surface, called also area, has length and breadth, but no thickness.

4. A solid has length, breadth, and thickness.

5. A right line is the shortest that can be drawn between two points.

6. The inclination of two lines meeting one another, or the opening between them, is called an angle. Thus, at B, Fig. 1, is an angle, formed by the meeting of the lines A B and B C.

Fig. 1.

A

7. If a right line CD, Fig. 2, fall upon another right line AB, so as to incline to neither side, but make the angles on each side equal, then those angles are called right angles; and the line CD is said to be

B

Fig. 2.

D

8. An obtuse angle is greater than a right angle; as A DE, Fig. 3.

9. An acute angle is less than a right angle; as E DB, Fig. 3.

Fig. 3.

NOTE.

When three letters are used to express an angle, the middle letter denotes the angular point.

10. A circle is a round figure bounded by a single line, in every part equally distant from some point, which is called the centre. Fig. 4.

11. The circumference or periphery of a circle is the bounding line; as A DEB, Fig. 4.

12. The radius of a circle is a line drawn from the centre to the circumference; as CB, Fig. 4. Therefore all radii of the same circle are equal.

13. The diameter of a circle is a right line drawn from one side of the circumference to the other, passing through the centre; and it divides the circle into two equal parts, called semicircles; as A B or DE, Fig. 5.

14. The circumference of every circle is supposed to be divided into 360 equal parts, called degrees; and each degree into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds ; and these into thirds, &c.

NOTE. Since all circles-are divided into the same number of degrees, a degree is not to be accounted a quantity of any determinate length, as so many inches or feet, &c. but is always to be reckoned as being the 360th part of the circumference of any circle, without regarding the size of the circle.

15. An arc of a circle is any part of the circumference; as BF or FD, Fig. 5; and is said to be an arc of as many

Fig. 6.

H

16. A chord is a right line drawn from one end of an arc to another, and is the measure of the arc; as H G is the chord of A the arc HIG, Fig. 6.

NOTE. The chord of an arc of 60 degrees is equal in length to the radius of the circle of which the arc is a part.

17. The segment of a circle is a part of a circle cut off by a chord; thus, the space comprehended between the arc HIG and the chord H G is called a segment. Fig. 6.

18. A sector of a circle is a space contained between two radii, and an arc less than a semicircle; as BCD, or AC D, Fig. 6.

19. The sine of an arc is a line drawn from one end of the arc, perpendicular to the radius or diameter drawn through the other end: or, it is half the chord of double the arc; thus, HL is the sine of the arc HB, Fig. 7.

20. The sines on the same diameter increase in length till they come to the cen-A tre, and so become the radius, after which they diminish. Hence it is plain that the sine of 90 degrees is the greatest possible sine, and is equal to the radius.

21. The versed sine of an arc is that part of the diameter or radius which is between the sine and the circumference; thus LB is the versed sine of the arc HB, Fig. 7.

22. The tangent of an arc is a right line touching the circumference, and drawn perpendicular to the diameter; and is terminated by a line drawn from the centre through the other end of the arc; thus, B K is the tangent of the arc B H, Fig. 7.

NOTE. The tangent of an arc of 45 degrees is equal in length to the radius of the circle of which the arc is a

23. The secant of an arc is a line drawn from the centre through one end of the arc till it meets the tangent; thus, CK is the secant of the arc BH, Fig. 7.

24. The complement of an arc is what the are wants of 90 degrees, or a quadrant: thus, HD is the complement of the arc B H, Fig. 7.

25. The supplement of an arc is what the arc wants of 180 degrecs, or a semicircle; thus ADH is the supplement of the arc BH, Fig. 7.

NOTE. It will be seen by reference to Fig. 7, that the sine of any arc is the same as that of its supplement. So, likewise, the tangent and secant of any arc are used also for its supplement.

26. The sine, tangent, or secant, of the complement of any arc, is called the co-sine, co-tangent, or co-secant of the arc; thus, F H is the sine, DI the tangent, and C I the secant of the arc DH; or they are the co-sine, co-tangent, and co-secant of the arc B H, Fig. 7.

[The terms sine, tangent, and secant, are abbreviated thus: sin., tan., and sec. So, likewise, co-sine, co-tangent, and cosecant, are written co-sin., co-tan., and co-sec.]

27. The measure of an angle is the arc of a circle contained between the two lines which form the angle, the angular point being the centre; thus, the angle H C B, Fig. 7, is measured by the arc BH; and is said to contain as many degrees as the arc does.

NOTE. An angle is esteemed greater or less, according to the opening of the lines which form it, or as the arc intercepted by those lines contains more or fewer degrees. Hence it may be observed, that the size of an angle does not depend at all upon the length of the including lines; for all arcs described on the same point, and intercepted by the same right lines, contain exactly the same number of degrees, whether the radius be longer or shorter. 28. The sine, tangent, or secant of an arc, is also the sine, tangent, or secant of the angle whose measure the arc is.

29. Parallel lines are such as are equal- A ly distant from each other; as A B and C D,

Fig. 8.

B