THEOREM XIX.

Similar polygons described in circles, are to each other, as the circles in which they are inscribed; or as the squares of the diameters of those circles. (Em. IV. 36.)

THEOREM XX.

All similar solids are to each other, as the cubes of their like dimensions. (Em. VI. 24.)

AN

EXPLANATION

OF THE

Principal Mathematical Characters.

=

THE sign or character (called equality) denotes that the respective quantities, between which it is placed, are equal; as 4 poles 22 yards = 1 chain = 100 links.

The sign+(called plus, or more) signifies that the numbers between which it is placed, are to be added together; as 9 + 6 (read 9 plus 6) = 15. Geometrical lines are generally represented by capital letters; then AB + CD, signifies that the line AB is to be added to the line CD.

The sign (called minus, or less) denotes that the quantity which it precedes, is to be subtracted; as 156 (read 15 minus 6) = 9. In geometrical lines also, AB CD, signifies that the line CD is to be subtracted from the line AB.

The sign x denotes that the numbers, between which it is placed, are to be multiplied together; as 5 x 3 (read 5 multiplied by 3) = 15.

The sign signifies division; as 153 (read 15 divided by 3) = 5. Numbers placed like a vulgar fraction, also denote division; the upper number being the dividend, and the lower the divisor; as 15 = 5. The signs: (called proportionals) denote proportionality; as 2: 5 :: 6 : 15, signifying that the number 2 bears the same proportion to 5, as 6 does to 15: or, in other words, as 2 is to 5, so is 6 to 15.

The sign (called vinculum) is used to connect several quantities together; as 9 + 3 | − 6 × 6 x 2 = 6 x 2 = 12.

2 = 12

The sign, placed above a quantity, represents the square of that quantity; as 5 + 3/2 = 8o = 8 × 8 64.

3

The sign, placed above a quantity, denotes the cube of that quantity; as 9+385 = 12 = 43 = 4 X 4 X 4 = 64.

2

8

The signor✔✅, placed before a quantity, denotes the square root of that quantity; as √9 x 4 = √36 = 6.

The sign, placed before a quantity, represents the cube root of that quantity; as 6 x 4 × 3

5

3

8/64 = 4.

8

PART II.

MENSURATION

OF

SUPERFICIES.

THE area of any plane figure is its superficial content, or the measurement of its surface, without any regard to thickness.

The dimensions of figures are taken in lineal measure. Sometimes they are taken in inches and tenths; sometimes in feet, inches, and parts; sometimes in feet, tenths, and hundredths; and sometimes in chains and links; and the area of any figure is estimated by the number of square inches, square feet, square chains, &c. contained in that figure.

Note 1. A measuring tape, (usually called a box and tape) divided into feet and inches on one side, and links on the other, is admirably calculated for taking dimensions. Tapes are of various lengths; but those of four poles, or twenty-two yards, are most useful.

2. In practice, the dimensions may generally be commodiously entered upon a rough sketch of the figure.

3. The following tables of lineal and square measures, ought to be well understood by the learner, before he proceeds further. A TABLE OF LINEAL MEASURE.

Note. Two yards make one fathom; and seven yards one rood of fencing or

PROBLEM I.

To find the area of a square.

RULE.

Multiply the length of one of its sides by itself, and the product will be the area.

Note 1. The side of asquare may be found by extracting the square root of its area.

2. When the area of a figure is found by duodecimals, it is generally said to be in feet, inches, parts, &c., but it is evident that it is in feet, twelfths, inches, &c.; because 144 square inches, and not 12, make a square foot. In this work, I have, however, eonformed to the general practice."

3. The learner should carefully examine, work over, and put down all the solutions given in this book, in order that he may the better comprehend the different Rules. The Definitions and Rules should also be committed to memory.

EXAMPLES.

1. What is the area of the square ABCD, the side of which is 8 feet 6 inches?

2. Required the area of a square floor, whose side

is 18 feet 9 inches.