2. Lay down a line that ranges S. W. b. W., making an angle of 56° 15', with the meridian line.

Draw the meridian line A S; and with the sweep of 60° describe the arc E F.

Set off 56° 15' from E to F; draw the line A F, and it will range S. W. b. W., as was required.

NOTE 1.-If the line had ranged S. E. b. E., the angle must have been set off from E to G; and A G would have been the direction of the line.

2. This Problem will be found useful to young Surveyors, in laying down the first line, the range of which should be taken in the field by a compass.

GEOMETRICAL THEOREMS,

THE DEMONSTRATIONS OF WHICH MAY BE SEEN IN THE ELEMENTS OF EUCLID, SIMPSON, AND EMERSON.

The greatest side of every triangle is opposite to the greatest angle. (Euc. I. 18. Simp. I. 13. Em. II. 4.)

THEOREM III.

A

E

Let the right line E F fall upon the parallel right lines A B, CD; the alternate angles A G H, GHD are equal to each other; and the exterior angle E G B is equal to the interior and opposite, upon the same side G HD; and the two interior angles B G H, G HD, upon the same side, are together equal to two right angles. (Euc. I. 29. Simp. I. 7. Em. I. 4.)

G

B

D

H

F

C

THEOREM IV.

A

Let A B C be a triangle, and let one of its sides B C be produced to D; the exterior angle A C D is equal to the two interior and opposite angles CAB, ABC; also the three interior angles of every triangle are together equal to two right angles. (Euc. I. 32. Simp. I. 9 & 10. Em. II. 1 & 2.)

THEOREM V.

Let the parallelograms A B C D, DBCE be upon the same base B C, and between the same parallels A E, B C; the parallelogram A B C D, is equal to the parallelogram D B C E. (Euc. I. 35. Simp. II.

B

D

2. Em. III. 6.)

THEOREM VI.

A D

Let the triangles A B C, DBC be upon the same base B C, and between the same parallels A D, B C; the triangle A B C is equal to the triangle D B C. (Euc. I. 37. Simp. II. 2. Em. II. 10.)

THEOREM VII.

Let A B C be a right-angled triangle, having the right angle B A C; the square of the side B C is equal to the sum of the squares of the sides A B, A C. (Euc. I. 47. Simp. II. 8. Em. II. 21.)

NOTE. Pythagoras, who was born about 2450 years ago, discovered this celebrated and useful Theorem; in consequence of which, it is said, he offered a hecatomb to the gods.

THEOREM VIII.

Let A B C be a circle, and B D C an angle at the centre, and B A C an angle at the circumference, which have the same arc B C for their base; the angle B D C is double of the angle B A C. (Euc. III. 20. Simp. III. 10. Em. IV. 12.)

A

B

C

THEOREM IX.

Let A B C be a semi-circle; then the angle A B C in that semi-circle, is a right angle. (Euc. III. 31. Simp. III. 13. Em. VI. 14.)

THEOREM X.

Let D E be drawn parallel to B C, one of the sides of the triangle A B C; then B D is to DA, as C E to E A. IV. 12. Em. II. 12.)

(Euc. VI. 2. Simp.

THEOREM XI.

In the preceding figure, D E being parallel to B C, the triangles A B C, ADE are equi-angular or similar; therefore A B is to B C, as AD to DE; and A B is to A C, as A D to A E. (Euc. VI. 4. Simp. IV. 12. Em. II. 13.)

THEOREM XII.

A

Let A B C be a right-angled triangle, having the right angle BA C; and from the point A let A D be drawn perpendicularly to the base B C; the triangles A B D, A D C are similar to the whole triangle A B C, and to each other. Also the perpendicular A D is a mean proportional between the segments of the base; and each of the sides is a mean proportional between the base and its segment adjacent to that side; therefore B D is to D A, as DA to DC; BC and B C is to CA, as CA to C D. Em. VI. 17.)

is to B A, as B A to B D;

(Euc. VI. 8. Simp. IV, 19.

THEOREM XIII.

B

A

D C

Let A B C, A D E be similar triangles, having the angle A common to both; then the triangle A B C is to the triangle A D E, as the square of B C to the square of D E. That is similar triangles are to one another in the duplicate ratio of their homologous sides. (Euc. VI. 19. Simp. IV. 24. Em. II. 18.)

D

E

THEOREM XIV.

In any triangle A B C, double the square of a line C D, drawn from the vertex to the middle of the base A B, together with double the square of half the base A D or B D, is equal to the sum of the squares of the other sides A C, B C. A (Simp. II. 11. Em. II. 28.)

THEOREM XV.

In any parallelogram A B C D, the sum of the squares of the two diagonals A C, B D, is equal to the sum of the squares of all the four sides of the parallelogram. (Simp. II. 12. Em. III.

9.)

A

THEOREM XVI.

All similar figures are in proportion to each other as the squares of their homologous sides. (Simp. IV. 26. Em. III. 20.)

THEOREM XVII.

The circumferences of circles, and the arcs and chords of similar segments, are in proportion to each other, as the radii or diameters of the circles. (Em. IV. 8 & 9.)

THEOREM XVIII.

Circles are to each other as the squares of their radii, diameters, or circumferences. (Em. IV. 35.)

THEOREM XIX.

Similar polygons described in circles are to each other, as the circles in which they are inscribed; or as the squares of the diameters of those circles. (Em. IV. 36.)

THEOREM XX.

All similar solids are to each other, as the cubes of their like dimensions. (Em. VI. 24.)

PART THE SECOND.

A DESCRIPTION OF THE CHAIN, CROSS-STAFF, OFFSETSTAFF, COMPASS, AND FIELD-BOOK; ALSO DIRECTIONS AND CAUTIONS TO YOUNG SURVEYORS, WHEN IN THE FIELD, ETC.

THE CHAIN.

LAND is commonly measured with a Chain, invented by Mr. Gunter, which is known by the name of "Gunter's Chain."

It is 4 poles, 22 yards, or 66 feet in length, and divided into 100 equal parts, called links; each link being 7.92 inches. At every tenth link from each end, is fixed a piece of brass, with notches or points; that at 10 links having one notch or point; at 20, two; at 30, three; and at 40, four points. At 50, or the middle, is a large, round, plain piece of brass.

The chain being thus marked, the links may be easily counted from either end; the mark at 90, 80, &c. being the same as that at 10, 20, &c. Part of the first link, at each end, is made into a large ring or bow, for the ease of holding it in the hand.

The chain should always exceed 22 yards, by an inch and half, or two inches; because, in surveying, it is almost impossible to go in a direct line, or to keep the chain perfectly stretched. Long arrows likewise keep the ends of the chain a considerable distance from the ground; the lines, consequently, will be made longer than they are in reality.

Chains, when new, are seldom a proper length; they ought always, therefore, to be examined; as should those, likewise, which are stretched by frequent use.

NOTE 1.—In folding up the chain, it is most expeditious to begin at the middle, and fold it up double. When you wish to unfold it, take both the handles in your left-hand, and the other part of the chain in your right; then throw it from you, taking care to keep hold of the handles. You must then adjust the links before

you proceed to measure.

2. Chains, which have three rings between each link, are much better than those which have only two; as they are not so apt to twist.

Chains thus made are sold by Jones and Co., Mathematical Instrument Makers, Holborn, London.