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What is Involution? Ans. It is finding a power, or a power of a root.

EXTRACTION OF THE SQUARE ROOT. What does any number multiplied into itself produce? Ans. A square.

What is it then to extract the square root? Ans. It is only to find what number, multiplied into itself will make the given number.

What is the square root of 64? Ans. 8.
Why? Ans. Because 8 times 8 are 64.

What is the proof of the square root then? Ans. Multiply the answer or root into itself.

What must the result be like? Ans. The given sum.

What is the square root of 144, and the proof?
What is the square root of 400 and the proof?
What is the square root of 100 and the proof!

A man desirous of making his kitchen garden, which is to contain 24 acres, or 400 rods, a complete square; what will be the length of the side?

A square lot of land is to contain 22} acres,or 3600 rods of ground, but for the sake of fruit, there is to be a smaller square within the larger, which is to contain 225 rods; what is the length of each square? Ans. 60 rods the outer, 15 rods the inner.

One hundred scholars are to be placed in a square roum, how many will that be on each side?

Suppose 40 boys should collect together to per. form some military evolutions, and should wish to march through the town in a solid phalanx, or square body, how many would the first rank consist of?

A General has 400 men, how many must he place in rank and file, to form them into a square?

A certain square pavement contains 1600 square stones, all of the same size; I demand how many are contained in one of its sides? Ans. 40.

EXTRACTION OF THE CUBE ROOT. What is the third power of any number called? Ans. A cube.

What is it to extract the cube root? Ans. It is only to find what number used as a factor three times will make the given sun or power; or it is that number which multip.ied into its square, will produce the given number.

What is the proof then? Ans. Multiply the answer or root into itself till it is taken as a factor 3 times.

What is the cube root of 8? Ans. 2.

Why, or proof? Ans. Because 2x2x2=is equal to 8.

What is the cube root of 27? Ans. 3.
Why? Ans. Because 3X3X3=27.
What is the cube root of 64? Ans. 4.
What is the proof?
What is the cube root of 1000? Ans. 10.
Why; or proof.
What is the cube root of 8000? Ans. 20.
What is the proof? Ans. 20 x 20 x 20=8000,

The solid contents of a square pile of wood are 216 feet: I demand the length and breadth of said pile? Ans. 6 feet.

What is the proof?

In Involution you recollect that a boy got by his knowledge of powers a square box filled with marbles, containing 1000 ; now how many will reach gross the bottom in a straight row, and how many each from the top to the bottom in a straight line.

You recollect also the sum in Involution about potatoes, the size, the bin, &c.- the bin held 40 bushels; now how many will reach from one side of the bottom to the other in a right line-or straight row.

In Involution John found a piece of gold which brought 80 dollars, that is 10 dollars for each solid inch; what may be the shape of the piece, and the length of each side.

What is the length of one side of a vessel, which contains one million of solid feet? Ans. 100 feet.

Why? Ans. Because 100 x 100 x 100=1,000,000.

ALLIGATION. What is Alligation? Ans. The mixture of fluids and medicine.

How many kinds does it consist of? Ans. 3.

What are they? Ans. Alligation Medial, , Alternate and Partial.

When do you employ Alligation Medial? Ans. When the quantities and prices of several things are, given to find the mean price of the mixture compound of those materials.

In the rule, what make the first term? Ans The whole composition.

What make the second? Ans. The whole value.

What make the third? Ans. Any part of the composition.

What will the fourth term be! Ans. The mean price.

Note.--The remaining rules in Alligation as they cannot be abridged, or simplified, may as well be learned from the Arithmetic of which this is an accompaniment. Some useful rules for finding the contents of super

ficies and solids. What is a straight line? Ans. The nearest distaner between two points.

If two lines are at equal distances from one another in every part, what are they called? Ans. Parallel lines.

If two lines are not at equal distances from each other in every part, what is the consequence, if they are continued from those parts that incline to each other? Ans. They must meet.

What is the place of meeting called? Ans. An angle as A in figure 1st.

What may the corner of any thing be called? Ans. An angle.

When a line falls directly on another line so as to make the angles on both sides equal; what is the line called? Ans. A perpendicular.

What are the angles on each side called? Ans. Right angles.

How many degrees do a right angle consist of? Ans. 90 as C in figure 3d.

What are all angles except right angles, called? Ans. Oblique.

How many kinds of oblique angles are there! Ans. Two.

What are they? Ans. Acute and Obtuse.
What is an acute angle? Ans. Less than a right.
What is an obtuse ? Ans. Greater than a right.

Of how many degrees does an acute angle con. sist of then? Ans. Less than 90.

How many in an oblique? Ans. More than 90.

Figure 1.

A SQUARE.

What is a square? Ans. A figure whose sides are all equal and parallel.

How many degrees in each angle of a square! Ans, 90.

What are the angles called then?

How many angles in all the angles of a square then?

What is the area of anything? Ans. The surface or space enclosed by any lines.

To find the area of a square, what is the rule? Ans. Multiply the side of the square into itself.

How many square feet of boards are contained in the floor of a room, which is 30 feet square? Ans. 900.

Suppose a square lot of land measures 40 rods on each side, how many acres does it contain? 40X40 =-1600-160=10 the answer.

Why divide by 160 in the above? Ans. Because the answer is square rods, and square rods must be divided by square rods, square feet by square feet, solid feet by solid feet, &c.

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What is a parallelogram? Ans. It is a long square, or it is a figure, that had its opposite sides equal, and parallel.

What is the rule for measuring a parallelogram? Ans. Multiply the length by the breadth.

A garden is 200 feet long, and ten feet wide; how many feet of ground are there in it? Ans. 2000 feet.

If a board be 10 inches wide and 15 feet long; how many feet in it? Ans. 1 foot.,

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