2. If the first term be 100, the common ratio 1.06, and the number of terms 5, what is the last term? Ans. 126.2477.

NOTE.-As the principles of arithmetical progression may be applied with advantage to the computation of annual interest, so may those of geometrical progression in computing compound interest. When thus applied the principal is the first term, the amount the last term, the number of regular intervals, at the end of which the interest is to be compounded, one less than the number of terms, and the amount of one dollar for one of those intervals the common ratio. To find the different powers of the ratio, the table on pages 132 and 133 may be used, the number in the column of years indicating the degree of the power; e. g., the 50th power of 1.03 is 5.58492686.

3. What is the amount of $100 for 50 years at 10% compound interest? Ans. $11739.09. 4. If a man beginning at the age of 21, at the end of each year puts $100 at compound interest, what will these sums amount to when he is 50 years old? Ans. $7363.98.

5. A gentleman offered for sale a lot of ten acres on the following terms: one mill for the first acre, one cent for the second, one dime for the third, and so on in geometrical progression. What was his price for the whole?

Ans. $1111111.111.

6. What is the sum of the series, T, Too, &c., or .333, &c., carried to infinity? Ans.. 7. What common fraction is equivalent to the repetend .7777, &c. ? Ans. 1.

8. At 12 o'clock the hour and minute hands of a clock are together. In what time will they be together again ?

SOLUTION.-When the minute hand has performed one entire revolution around the face of the clock, the hour hand will be 1⁄2 of a revolution in advance. When the minute hand shall have gone over this, the hour hand will still be I of that twelfth in advance, or of an entire revolution. When the minute hand shall have reached that point, the hour hand will be of in advance, and so the comparison of their relative position may be supposed to be made an infinite number of times. It is evident that for the minute hand to overtake the hour hand, it must perform as many revolutions (and hence take as many hours)

as would be the sum of the series 1, 72, 747, 7737, &c., continued to infinity equal to 1 hours. With the above reasoning one might almost believe that the hour hand would always be ahead, but as a matter of fact we know that the minute hand does overtake and pass the hour hand, and therefore at some point the distance between the two must be nothing. Farthermore, as the series above represents the successive distances apart in their actual progress, we have from this case conclusive proof that the last term of an infinite decreasing geometrical series is absolutely nothing.

9. If an ivory ball is let fall upon a marble slab, from a height of 10 feet, and it rebounds 9 feet, falling again it rebounds 8.1 feet, and so continues always rebounding of the distance through which it fell last, will it ever come to rest, and if so, through what space will it have passed?

Ans. It would pass through 190 feet.

10. If the banking law of Illinois allows the State Auditor to issue to any banker depositing State Stocks, 90 per cent. of the par value of those stocks in circulating bank notes, without farther restriction, what is the amount of Stocks a banker could so put on deposit with only $10000 Cash Capital, if he continue to re-invest the bank notes for other Stocks both at par, until he should have nothing to re-invest? If the Stocks draw 6% interest, what dividend does he realize on his capital? Ans. to the first $100.000.

ART. 174. A point has neither length, breadth, nor thickness, but position only.

A line has length without breadth or thickness, and may be straight or curved.

A surface has length and breadth without thickness, and may be plain or curved.

A solid has length, breadth, and thickness.

An angle is the divergence of two straight lines from a common point. When the divergence is equal to that made

by a straight line and one perpendicular to it, it is called a right angle, and its measure is 90 degrees (90°). A less divergence forms an acute angle, and a greater an obtuse angle.

The area of a figure is its quantity of surface, and is measured by the product of the linear dimensions of length and breadth, which will give the number of square units of the same denomination covering an equivalent surface.

REMARK.-The only difficulty then in computing the area of any figure is to find the linear dimensions of its average length and breadth, or those of another figure known to be of equal area. Take for example the "quadrature of the circle." It can easily be proven that the area of a circle is equal to the area of a rectilinear figure, with a length equal to the circumference of the circle, and a breadth equal to half the radius; but as our system of notation will not express the exact length of the radius for a given circumference, nor the exact length of the circumference for a given radius, the problem will not admit of an exact solution, though the approximation may be carried to an indefinite extent.

The solidity or volume of a solid or body is the quantity of space which it occupies, and is measured by the product of the three linear dimensions of length, breadth, and thickness, which will give the number of cubic units of the same denomination occupying an equivalent space.

A rectilinear figure, or polygon, is a plane figure bounded by straight lines. A polygon of three sides is called a triangle, of four sides a quadrilateral, of five a pentagon, of six a hexagon, and so on.

A regular polygon is one whose sides and angles are equal. A trapezium is a quadrilateral which has no two sides parallel.

A trapezoid is a quadrilateral which has only two sides parallel.

A parallelogram is a quadrilateral whose opposite sides are equal and parallel.

The altitude of a parallelogram or trapezoid is the perpendicular distance between the parallel sides.

A rectangle is a right-angled parallelogram.

A square is an equilateral rectangle.

A rhombus is an equilateral parallelogram with only its opposite angles equal.

A rhomboid is a parallelogram neither equilateral nor equiangular.

Similar figures are those whose corresponding angles are equal, and the sides about the equal angles proportional.

The areas of similar figures are to each other as the squares of their corresponding linear dimensions, and the volumes of similar solids are to each other as the cubes of their corresponding linear dimensions.

TRIANGLES.

ART. 175. In computing the area of a triangle, either side may be assumed as the base, and the altitude will be the perpendicular let fall from the vertex of the angle opposite upon the base, or base produced if necessary.

To find the area of a triangle.

RULE.-Multiply the base by half the altitude, and the product will be the area; or

Take half the sum of the three sides, and from this subtract each side separately; then multiply together the half sum and the three remainders, and the square root of the product will be the area.

Examples.

1. How many square yards in a piece of ground of triangular shape, one side measuring 50 yards, and the shortest distance from this side to the opposite angle being 24 yards? Ans. 600 sq. yds.

2. The three sides of a triangle measure respectively 10, 12, and 14 feet; what is the area? Ans. 58.7878 sq. ft.

3. How much greater would be the area if we double the linear dimensions in the last example ! Ans. Four times.

4. What should be the dimensions of a triangle similar to the one proposed in example 1, to make the area 5400 sq. yards instead of 600 ? Ans. The base 150 yds. The altitude 72 yds.

5. If one side of a field containing 50 acres is 50 rods,

what must be the length of the corresponding side of a field of similar shape to contain 112 acres ?

Ans. 75 rods. 6. The area of a certain triangular field is 33 acres, and one of its sides is 37 rods long; what is the length of a perpendicular from the opposite corner ? Ans. 32 rods. 7. What is the side of a square containing the same area as a triangle whose base is 36.1 feet, and altitude 5 feet? Ans. 91 feet.

QUADRILATERALS, PENTAGONS, &c. ART. 176. (1.) To find the area of any quadrilateral having two sides parallel.

RULE.-Multiply half the sum of the two parallel sides by the altitude, or perpendicular distance between those sides, and the product will be the area.

NOTE. This rule is equally applicable to the square, rectangle, rhombus, rhomboid, and trapezoid. If the parallel sides are equal, the half sum would be equal to one of them. (2.) To find the area of a regular polygon.

RULE.-Multiply the sum of the sides or perimeter by half the perpendicular let fall from the center upon one of its sides. Or,

Multiply the square of one of the sides by the appropriate number as given in the following

(3.) To find the area of an irregular polygon of four or more sides.

RULE.-Divide the figure into triangles by diagonals connecting some one angular point with each of the others; compute the area of each triangle, and their sum will be the area required.