The Side AC will be found by CASE I. to be nearest 253.

Note. If the given Angle be Obtuse the Angle sought will be Acute; but if the given Angle be Acute, and opposite a given lesser Side, then the Angle found by the operation may be either Obtuse or Acute. It ought therefore be mentioned which it is, by the conditions of the question.

By Natural Sines.

As the Side opposite the given Angle; Is to the Nat. Sine of that Angle; So is the other given Side; To the Nat. Sine of its opposite Angle.

One given Side 200; Nat. Sine of 46° 30′, its opposite Angle, 0.72537; the other given side 240.

As 200 0.72537: 240 0.87044 50° 30′.

CASE III.

Two Sides and their contained Angle given, to find the other Angles and Side. Fig. 49.

The solution of this CASE depends on the following PROPOSITION.

In every Plane Triangle, As the Sum of any two Sides; Is to their Difference; So is the Tangent of half the Sum of the two opposite Angles; To the Tangent of half the Difference between them. Add this half difference to half the Sum of the Angles and you will have the greater Angle; and substract the half Difference

In the Triangle ABC, given the side AB, 240, the Side AC 180, and the Angle at A 36° 40′ to find the other Angles and Side.

Side AB

The given Angle BAC 36° 40′, subtracted from

180°, leaves 143° 20′ the Sum of the other two Angles;

the half of which is 71° 40'.

As the Sum of two Sides, 420

: Their Difference, 60

:: Tangent half unknown Ang. 71° 40′

: Tangent half Difference, 23° 20′

The half sum of the two unknown Angles
The half difference between them,

Add, gives the greater Angle ACB

Subtract, gives the lesser Angle ABC

2.62325

1.77815

10.47969

12 25784 2.62325

9.63459

71°40'

23 20

95 00

48 20

The Side BC may be found by CASE I or II.

CASE IV.

The three Sides given to find the Angles. Fig. 50. The solution of this CASE depends on the following PROPOSITION.

In every Plane Triangle, As the longest side; Is to the Sum of the other two Sides; So is the Difference between those two Sides; To the Difference between the Segments of the longest Side, made by a Perpendicular let fall from the Angle opposite that Side.

Half the Difference between these Segments, added

length of the longest Side, will give the greatest Segment; and this half Difference subtracted from the half Sum will be the lesser Segment. The Triangle being thus divided becomes two Right Angled Triangles, in which the Hypothenuse and one Leg are given to find the Angles.

In the Triangle ABC, given the Side AB 105, the Side AC 85, and the Side BC 50; to find the Angles.

As the longest Side AB, 105

: Sum of the other two Sides, 135
:: Difference between those Sides, 35

Difference

35

2.02119

2.13033

1.54407

3.67440

2.02119

Thus the Triangle is divided into two Right Angled Triangles, ADC and BDC; in each of which the Hypothenuse and one Leg are given to find the Angles.

To find the Angle DCB.
1.92942 As Hyp. BC, 50
10.00000 : Radius

1.69897 10.00000

1.87506 :: Seg. BD, 30

The Angle DCA 61° 55′ subtracted from 90° leaves the Angle CAD 28° 5′

The Angle DCB 36° 50′ subtracted from 90' leaves the Angle CBD 53° 10'

The Angle DCA 61° 55′ added to the Angle DCB 36° 50′ gives the Angle ACB 98° 45′

This Case may also be solved according to the following PROPOSITION.

In every plane Triangle, As the Product of any two Sides containing a required Angle; is to the Product of half the Sum of the three Sides, and the Difference between that half Sum and the Side opposite the Angle required; So is the Square of Radius; To the Square of the Co-Sine of half the Angle required.

GIMETRY.

Those who make themselves well acquainted with TRIGONOMETRY will find its application easy to many useful purposes, particularly to the mensuration of Heights and Distances; called ALTIMETRY and LONThese are here omitted because, as this work is designed principally to teach the Art of common FIELD SURVEYING, it was thought improper to swell its size, and consequently increase its price, by inserting any thing not particularly connected with that Art.

It is recommended to those who design to be Surveyors to study TRIGONOMETRY thoroughly; for though a common Field may be measured without an acquaintance with that Science, yet many cases will occur in practice where a knowledge of it will be found very beneficial; particularly in dividing Land, and ascertaining the boundaries of old Surveys. Indeed no one who is ignorant of TRIGONOMETRY, can be an accomplished Surveyor.

SURVEYING.

SURVEYING is the Art of measuring, laying out and dividing Land.

PART I.

MEASURING LAND.

THE most common measure for Land is the Acre; which contains 160 Square Rods, Poles or Perches; or 4 Square Roods, each containing 40 Square Rods.

The instrument most in use, for measuring the Sides of Fields, is GUNTER'S Chain, which is in length 4 Rods or 66 Feet, and is divided into 100 equal parts, called Links, each containing 7 Inches and 92 Hundredths. Consequently, 1 Square Chain contains 16 Square Rods, and 10 Square Chains make 1 Acre.

In small Fields, or where the Land is uneven, as is the case with a great part of the Land in New-England, it is better to use a Chain of only two Rods in length;