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As S. of CDA

Is to S. of CAD
So is side AC

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114°,00′ 0.03927) As S. of CDA 114° - 0·03927 22°,00' 9.57358 Is to S. of ACD 44° - 9-84177 489.5 2.68973 So is side AC

To perpend. hht. CD200-7 2.30258 To side AD

489.5-2.68973

372.2-2.57077

To find the height of the mountain and object together; we have the right angled triangle ACE, in which are given the hypothenuse AC 489-5, angle CAE 46°, and the angle ACE 44°, whence, by Problem 24. of right angled Trigonometry, we have these propor

tions.

As radius

Is to S. of CAE
So is hypoth. AC

To per. hht. CE

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352.1 2.54666

To AE

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If you subtract CD from CE, you will have the height of the hill

151.4.

Any figure in Navigation, or Mensuration of Heights and Distances, may be measured Geometrically, as directed in the foregoing Problems of Trigonometry.

MENSURATION

OF SUPERFICIES AND SOLIDS.

SECTION I. OF SUPERFICIES.

SUPERFICIES, or surfaces, are measured by the superficial inch, foot, yard, &c. according to the measures peculiar to different artists. The superficial inch, foot, &c. is one inch, foot, &c. in length and breadth; and, because 12 inches make one foot of Long Measure, therefore, 12x12=144 inches make 1 superficial foot, 3×3=9 feet, a yard, &c.

The superficial content of every surface is found by the proper rule of its figure, whether square, triangle, polygon, or circle.

ARTICLE I. To measure a Square, having equal sides.

RULE. Multiply the side of the square into itself, and the product will be the area or superficial content, of the same name with the denomination taken, either in inches, feet, or yards, respectively. Let ABCD represent a square, whose side is 12 inches or 12 feet.

thus,

Multiply the side 12 by itself,

B

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By the Sliding Rule.

D

Set 1 to the length on B, then, find the breadth on A, and op posite to this on B, you will have the content,

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By Gunter's Scale.

Extend the dividers from 1, on the line of numbers, to the length; that distance, laid the same way from the breadth, will point out the

answer.

ART. 2. To measure a Parallelogram, or long Square. RULE.-Multiply the length by the breadth, and the product will be the area, or superficial content.

Let ABCD represent a parallelogram, Ai whose length is 16 feet, and breadth, 12 feet. Multiply 16 by 12.

Length 16

Breadth 12

192 area.

D

The content of this figure is found on the sliding rule and scale, as the former.

ART. 3. When the breadth of a Superficies is given, to find how much in length will make a square foot, yard, &c.

RULE. As the breadth is to a foot, yard, &c. so is a foot, yard, &c. to the length required to make a foot, yard, &c. Or divide 144 by the breadth, and the quotient will be the length required.

How much, in length, of a board 2 feet wide, will make a square foot?

In. br. In. leng. In. br. In. leng.
As 30 : 12 :: 12 : 4.8

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Breadth = 30)144(4-8 inches, Ans.

ART. 4. To measure a Rhombus.

Definition. A rhombus is a figure with four equal sides, in the form of a diamond on cards, having two angles greater and two less, than the angles of a square: the former are called obtuse angles, and the latter, acute, or sharp, angles.

RULE.-Multiply the side by the length of a perpendicular, let fall from one of the obtuse angles to the side opposite such angle, Let ABCD represent a rhombus,

each of whose sides is 16 feet: A per- A
pendicular let fall from the obtuse, an-
gle, at B, on the side DC, will inter-
sect it in the point E, so will BE be 12
feet; and this being multiplied into the
given side, the product will be the area
of the rhombus.

D

Side = 16
Per. = 12

192 area.

By the Sliding Rule.

Set 1 on A to the length on B; find the perpendicular height on A, against which on B is the content.

By Gunter.

The extent from 1 to the perpendicular height will reach from the length to the content.

ART. 5. To find the Area of a Rhomboides.

Definition. A rhomboides is a figure, whose opposite sides and op posite angles are equal

RULE. Multiply one of the longest sides by the perpendicular let fall from one of the obtuse angles on one of the longest sides. Let BCD represent a rhomboides ;

the longest sides AB and CD being 16.5

feet, and the perpendicular AE, 9.7 feet.

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RULE. If it be a right angled triangle, multiply the base by half the perpendicular, or half the base by the perpendicular, and the product will be the area: but if it be an oblique angled triangle, (whether obtuse, or acute) multiply half the base by the length of the perpendicular let fall on the base from the angle opposite to it, and the product will be the area. The longest side of a triangle is usually called the base, except in a right angled triangle, where the longest of the two legs, which include the right angle, is called the

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Set 1 on A to the length of the base on B, and opposite to half the length of the perpendicular, on A, you will have the content on B. By Gunter.

The extent from 1 to half the length of the perpendicular will reach from the length of the base to the content.

In this place it may be proper to instruct the learner in one of the properties of a right angled triangle: viz. That the square of the longest side of a right angled triangle, usually called the hypothenuse, is equal to the sum of the squares of the two other sides, usually called the legs; which is of great use, for by this mean, any two sides of a right angled triangle being given, the other may be found by common Arithmetick. Thus, in the right angled triangle ABC, the base AC and perpendicular BC being given, the hypothenue AB may be found by extracting the square root of the sum of the squares of the base and perpendicular.

Perp. 12.6

=

353 44 square of the base. 158.76 square of the perp.

512.20(22.63 hypothenuse.

Base 18.8

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4

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And, if the hypothenuse and one of the legs be given, the other may be found by subtracting the square of the given leg from the square of the hypothenuse.

There are some numbers, the sum of whose squares make a perfect square, of which sort are 3 and 4, whose squares, being added together, make 25, which is the square of 5: therefore, if the base of a triangle be 4, and the perpendicular 3, the hypothenuse will be 5; and if any of these numbers be multiplied by any other number, those products will be the sides of right angled triangles, as 6, 8, 10, and 15, 20, 25, &c. Thus artificers, when they set off the corner of a building, usually measure 6 feet on one side, and 8 feet on the other, then laying a 10 feet pole across, it makes the corner a true right angle.

ART.

ART. 7. There is another method of finding the area of triangles, the three sides being given.

RULE. Add the three sides together, then take the half of that sum, and out of it subtract each side severally; and multiply the half of the sum and these remainders continually, and the square root of this product will be the area of the triangle.

In the oblique triangle ABC, the base AC is given 15-6, the side AB is 10-4, and the side BC is 9.2, to find the area.

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ART. 3. To measure a Trapezium. 1

Definition. A trapezium is an irregular figure of four unequal sides, and unequal angles.

RULE.-Draw a diagonal line from one of the angles to the opposite angle, as AC; and then will the trapezium be divided into two triangles, of which the diagonal is the common base: then, letting fall perpendiculars from the other opposite angles on the diagonal, add those perpendiculars together, and multiply half that sum into the diagonal, or half of the diagonal into the sum of the perpendiculars, and that product will be the area of the trapezium.

In the trapezium ABCD, the diagonal AC is 24, the perpendicular DE 6, and the perpendicular BF 10. The sum of the perpendiculars is 16, whose half is 8, which being multiplied into 24,` will give the area.

A

E

D

B

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