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213. To extract the cube root of a vulgar fraction, if the numerator and denominator of the fraction be perfect cubes, we may find the cube root of each separately, and the answer will thus be obtained as a vulgar fraction; if not, first reduce the fraction to a decimal, or to a whole number and decimal, and then find the root of the resulting number. The answer will thus be obtained either as a decimal or as whole number and decimal; or we may multiply both numerator and denominater by such a number.

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2. 48228°544; 27054°036008; 12°977875; 27054036008; 21936532.

3.

065939264; 023639903; 000030664297; 001030301; 007645373.

4. 4 to three places of decimals; 7 to two places of decimals; & to two decimal places,

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6. Extract the cube root of 3 to six places, and also the cube root of 6, and show how the cube root of 2, of 12, and of 18, may be obtained from these.

7. Extract the cube root of oo90009009; and thence obtain the cube root of that cube root, and thence again the cube root of the last cube root to six and four places of decimals respectively.

ON FINDING THE VALUE OF FORMULA.

The following examples are intended to show the manner in which formulæ given in an algebraic appearance are to be treated.

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Here D2 (D squared) has to be multiplied by V, and the product divided by 6000. Although no sign is written between D2 and V, in all such cases the sign of multiplication is implied.

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Here the product of C and K, has to be subtracted from the product of C, and K, and the remainder divided by the difference of K, and K.

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PAV

Supposing the values attributed to v, p, A, V, ƒ and a to be the same, and it is required to find a. By attentively reading Nos. 191 and 192, page 140, it will be seen that v = fa means vfa=pA V, and by transposing, or in this case, dividing both sides of the equation PAV by vf, it becomes PAV or a = which in words means this:-The product of the v f quantities referred to as p A V, divided by the product or result produced by multiplication of the quantities referred to as vƒ, will give a as required. Thus,

of

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Similarly, the values of the other single letters are obtained in the same way.

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First find the quantity within the brackets-it is 25-subtract this from 12.

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21952 -2744
25

= 12 19998 = 12 X 768°32 = 9219'84 Ans.

; where f=4936, D = 41, H=64, and S = 32. Ans. 85*5.

*This was the Admiralty formula for N.H.P. for paddle engines.

A A

USEFUL RULES IN MENSURATION.

MENSURATION OF SUPERFICIES.*

Def.-A Parallelogram is a four-sided figure, of which the opposite sides are parallel.

Def.-A Square is a four-sided figure which has all its sides equal, and all its angles right-angles.

Def.-A Rectangle (or oblong) is a four-sided figure which has its opposite sides equal, and all the angles right-angles.

Def. The straight line joining two opposite angles of a quadrilateral is called a Diagonal.

Def. The Area of any plane figure is the measure of the space contained within its extremes or bounds, without any regard to thickness. The area or the content of the plane figure is estimated by the number of little squares that may be contained in it, the side of these little measuring squares being an inch, a foot, a yard, or any other fixed quantity. And hence the area or content is said to be so many square inches, or square feet, or square yards, &c.

Thus, if the figure to be measured be the rectangle ABCD, and the little square E, whose side is one inch, is the measuring unit proposed; then, as often as the said little square is contained in the rectangle, so many square units the rectangle is said to contain, which in this case is 15.

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214. To find the area of the square or any rectangular figure.

RULE LXIII.

Multiply the length by the breadth; the result is the required area.

NOTE. The dimensions must be of the same denominations, such as both feet or both inches, &c.; and if the product be inches, divide by 144 for feet, and if feet by 9 for square yards, &c.

Measurement of Surfaces.-The dimensions employed for areas of surfaces naturally follow from a length of the side of a square measured in linear feet and inches, and this brings us at once to square feet and square inches for surfaces or figures of two dimensions. Circles are greatly employed in the forms of machinery, because turning is the cheapest and easiest class of machine work. Boring a cylinder and turning a piston, for example, give great facilities for fitting the two together in a cheap and ready manner. All calculations as to the horse-power of an engine or the strength of its parts involve a knowledge of the areas of circles.

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Since the AREA = the LENGTH × the BREADTH, it follows that if any two of these be given, the third may be found as follows:

:

1. The Area may be found by multiplying the Length by the Breadth.

2.

The Length may be found by dividing the Area by the Breadth.

3. The Breadth may be found by dividing the Area by the Length.

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Ex. 3. How many yards of carpet 30 in. wide will be required for a room 28 feet long by 18 feet wide.

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We want to know how long a piece of carpet must be to cover this area supposing the carpet to be 30 inches wide; in other words, we have the area given—viz., 74592 sq. in.— and the width given—viz., 30 inches—and from these we find the length, for

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207 feet 2 inch = 69 yards 2 + inches. Answer.

The area of a parallelogram which is not right-angled is easily obtained from this consideration:-That we can convert it into a rectangle by cutting off a triangle at one end and putting it on at the other. Its area is thus seen to be the length multiplied into the breadth measured perpendicularly, or, as it is more commonly stated,

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Def.-A Rhombus is a four-sided figure which has all its sides equal, but its angles are not right-angles.

Def.-A Rhomboid is a four-sided figure which has its opposite sides equal, but its angles are not right-angles.

NOTE. It will be observed that the rhombus and rhomboid differ from the square and rectangle in one respect, viz., that their angles are not right-angles. They are, indeed, like squares and rectangles that have been forcibly wrenched (like a loose slate frame), from their true form. The rhombus being a distorted square, and the rhomboid a distorted rectangle.

These figures, because their opposite sides are parallel, are frequently called parallelograms, although that name really includes the square and rectangle.

EXAMPLE.

Ex. A rhombus is 12.4 feet long, and its perpendicular breadth is 109 feet: find the area Area = length x perpendicular breadth = 12'4 X 10'9 = 135°16 sq. feet.

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3. Required the area of a rectangle 273 inches long by 14 inches broad.

4.

A board is 7 inches broad by 3 feet 5 inches long: required its surface.

5.

The length of a parallelogram is 12 feet and its breadth 8 feet: what is its area.

6. A rectangle is to contain 133 square inches: what must be its length when the breadth is 7 inches.

7.

What length must be cut off a rectangular board whose breadth is 9 inches to make a square yard.

8. What is the area of a rhombus whose side is 7 ft. 6. in., and perpendicular height 3 ft. 4 in. ?

9.

What is the area of a rhombus whose length is 3 yards, and perpendicular height 2 ft. 3 in.?

215. To find the area of a triangle.

The easiest way by which to find the area of a triangle is by measuring the base and perpendicular height. In the accompanying figure let us suppose that we know the length of BC and A E. Now, if we multiply these together their product gives the area of the enclosing rectangle DBCF, which we know to be exactly double of the triangle ABC (Euc. I, 41). The area of which can, therefore, readily be found by dividing by 2.

RULE LXIV.

D

A

F

B

E

Multiply its base and height together, and half the product will be the area.

To find the area of a triangle, the base of which is b and perpendicular p.

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The base of a triangle is 76.5 feet, and perpendicular 92.2 feet, what is its area?

Area 765 X 92'2 ÷ 2 = 3526·65 square feet.

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