2. Lay down a line that ranges S.W. b. W., making an angle of 26° 15' with the meridian line.

Draw the meridian line AS; and with the chord of 60° describe the arc EF.

Set off 56° 15' from E to F; draw the line AF, 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 AG 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 WORKS OF EUCLID, SIMPSON, AND EMERSON.

THEOREM I.

If two straight lines AB and CD cut each c other in the point E, the angle AEC will be equal to the angle DEB, and CEB to AED A (Euclid, i. 15; Simpson, i. 3; Emerson, i. 2).

THEOREM II.

B

E

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

THEOREM III.

Let the right line EF fall upon the parallel right lines AB and CD; the

THEOREM IV.

A

Let ABC be a triangle, and let one of its sides BC be produced to D ; the exterior angle ACD is equal to the two interior and opposite angles CAB and 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).

B

Ꭰ

In the preceding figure, DE being parallel to BC, the triangles ABC and ADE are similar; therefore AB is to BC as AD to DE; and AB is to AC as AD to AE (Euc. vi. 4; Simp. iv. 12; Em. ii. 13).

THEOREM XII.

A

Let ABC be a right-angled triangle, having the right angle BAC ; and from the point A let AD be drawn perpendicularly to the base BC; the triangles ABD and ADC are similar to the whole triangle ABC, and to each other. Also the perpendicular AD is a mean proportional between the segments of the base; and each of the sides is a mean proportional between B. the base and its segment adjacent to that

C

D

side; therefore BD is to DA as DA to DC; BC is to BA as BA to BD; and BC is to CA as CA to CD (Euc. vi. 8; Simp. iv. 19; Em. vi. 17).

THEOREM XIII.

Let ABC and ADE be similar triangles, having the angle A common to both; then the triangle ABC is to the triangle ADE as the square of BC to the square of DE. That is, similar triangles are to one another in the duplicate ratio of their homologous sides (Euc. vi. 19; Simp. c iv. 24; Em. ii. 18).

D

B

C

THEOREM XIV.

In any triangle ABC, double the square of a line CD, drawn from the vertex to the middle of the base AB, together with double the square of half the base AD or BD, is equal to the sum of the squares of the other sides AC and BC (Simp. ii. 11; Em. ii. 28).

THEOREM XV.

In any parallelogram ABCD, the sum of the squares of the two diagonals AC and BD is equal to the sum of the squares of all the four sides of the parallelogram (Simp. ii. 12; Em. iii. 9).

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 II.

A DESCRIPTION OF THE CHAIN, CROSS-STAFF, OFFSET-STAFF, COMPASS, AND FIELD-BOOK; ALSO DIRECTIONS TO YOUNG SURVEYORS WHEN IN THE FIELD; &c.

THE CHAIN.

LAND is commonly measured with a Chain, invented by Mr Gunter, 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.