7. A merchant bought 80 yards of broadcloth, at 31 dollars per yard, and paid in coffee at 10 pence per pound, New York currency, how many pounds did it require? Ans. 2496 pounds.

8. Sold 24 hundred weight of hops at 4 pence per pound, and took pay in sugar at 6 pence per pound; how many pounds did I require? Ans. 16 cwt.

9. Sold 5 hundred weight 1 quarter of sugar at 8 dollars 50 cents per hundred weight, and received for pay 21 yards of cloth; what was the cloth per yard?

Ans. $1,12.

All the examples thus far, except No. 2, can be solved very briefly by Art. 24. No. 2 is also very brief, but

not in that form.

10. Q has coffee worth 16 cents per pound, but in barter, raised it to 18 cents; B has broadcloth worth 4 dollars 64 cents per yard; what must B raise his cloth to, so as to make a fair barter with Q? Ans. $5,22.

11. P and Q barter; P has Irish linen, worth 3 shillings 7 pence, cash, but, in barter, he will have 3 shillings 10 pence; Q delivers him broadcloth at 1 pound 16 shillings 6 pence per yard, worth only 1 pound 13 shillings; how much linen does P give Q for 138 yards of broadcloth, and which gains by the bargain? and how much in dollars and cents, the whole being in Irish currency? Ans. P gives 1314 yards; Q gains $33+, because his cloth is proportionally too high.

12. A and B barter; A has 150 galions of brandy, at 7 shillings 6 pence per gallon, cash, but in barter he will have 8 shillings; B has linen at 3 shillings 6 pence per yard, ready money; how much must it be per yard, to meet A's bartering price, and how many yards are equal to A's brandy?

Ans. Barter price, 3 s. 8g d.; B must give A 3213 yards, nearly.

SECTION V.

MENSURATION.*

(ART. 118.) MENSURATION is finding, or defining, the numerical contents of surfaces and solids, and its foundation is the science of geometry; but its every-day application to the mechanical business of life, demands a place for its practical rules in every arithmetic. Some of these rules are obvious, from inspection: others must be taken on trust, until the pupil studies geometry, from whence they are drawn; we shall, however, give some of the common illustrations. All surfaces, of whatever figure, are referred to squares for measurement, and the unit may be taken at pleasure; it may be an inch, a foot, a yard, a rod, a mile, &c., according as convenience and common sense may dictate. The following figures will render the truth of several rules apparent:

Fig. 1. A square.

Fig. 2. A right angled parallelogram.

1 2 3 4 5 6 7

The first, a square, the side of which is 3 units in length; and it is apparent to the eye, that the whole square contains 9 square units; and, considering the 3

* Most writers on arithmetic have placed mensuration after the square and cube roots; but we think it an injudicious arrangement, as, in the operations of finding roots, mensuration is almost universally used.

units in a side as feet, the whole square is one square yard, containing 9 square feet, agreeably to the dictation of the square measure table.

By figure 2nd, we perceive that the number of little squares it contains, is equal to the units in length, multiplied by the units in breadth, which must be true in every case.

(ART. 119.) If the sides are not at right angles, as in figure 3d, we must multiply its length by its perpendicular breadth, as it is proved in geometry, that the figure ABCG the figure BCDE.

Fig. 3. A right and an oblique-angled parallelogram.

A triangle is equal to half of a parallelogram, of the same base and altitude, as is obvious, from inspecting figure 4, AC the base, and BE the altitude.

A plain figure having 4 sides, and 2 of them parallel and unequal, is called a trapezoid. It is rigidly proved, in geometry, that the area of such a figure is found by multiplying its perpendicular altitude EF, by a medium base between AD and BC. This truth is apparent, from the inspection of Fig. 5.

(ART. 120.) When a circle is inscribed in a square, it has been ascertained by geometry, that the area or surface of the circle occupies ,7854 (as near as can be expressed with 4 decimals,) of the whole square. But the square is the square of the diameter of the circle, as appears from figure 6.

Therefore, to find the area of a circle, we may square the diameter, and multiply it by the decimal,7854.

It is also shown in geometry, that the circumference of a circle is 3,1416, nearly, when the diameter is 1; and when the diameter is 2, the circumference will be double,

Fig. 6.

D

and so on in proportion: ten times in diameter giving ten times for circumference, &c. &c.

Again: in figure 6, from the centre C, we can draw radii CD, CG, as near each other as we please; and if we draw them extremely near each other, the small part of the circumference GD, may be considered a straight line, and the sector CDG, a triangle; but the area of this triangle is equal to CD, multiplied by 1⁄2 of GD. But the whole circle can be filled up with such triangles: hence the area of the whole circle is equal to of its diameter, multiplied by 1⁄2 of all the GD's, or of the whole circumference. Squares, circles, and all figures of a similar shape to one another, are, in area, as the square, of their like dimensions. That is, a circle of double diameter to another, is 4 times in surface, of 3 times in diameter, 9 times in surface, &c. &c. This may be brought to the mind by inspecting figure 1. If we call the whole square in figure 1 an unit, then the square of is; that is, to square a fraction, square its numerator and denominator: geometrical figures and numericals agree.

3

RECAPITULATION.

To find the surface or area of a square.

RULE 1. Multiply its equal length and breadth together.

To find the area of a parallelogram.

RULE 2. Multiply length and breadth together, (taking care always to measure them at right angles.) To find the area of a triangle.

RULE 3. When the base and altitude are given, multiply one by half the other.

When not a right angle, and the altitude not known, but the three sides given, the following is the rule which the mere arithmetician must take on trust.

RULE 4. Add all the three sides together, and take half that sum. Next, subtract each side severally from the said half sum, obtaining three remainders. Then multiply the said half sum and those three remainders all together, and extract the square root of the last product, for the area of the triangle.

To find the area of a trapezoid, or a figure of 4 sides, and only two opposite sides parallel.

RULE 5. Add the two parallel sides together, and take half the sum: this will be the mean or average width. Multiply this mean width by the perpendicular distance between the sides.

To find the area of a circle.

RULE 6. Square the diameter, and multiply that square by the decimal ,7854.

Or, when most convenient, Multiply half the diameter by half the circumference.

[NOTE. The circumference to the diameter of any circle, is found to be very nearly in the proportion of 22 to 7: a more accurate ratio is 3,1416 to 1.]

To find the surface of a globe or sphere.

RULE 7. Multiply the circumference and diameter together; or, multiply the square of the circumference by 3,1416.