5. Find a fourth proportional to .05764, .7186, and .34721, by logarithms.

Ans. 4.328681 6. Find a fourth proportional to 12.687, 14.065, and 100.979, by logarithms.

Ans. 112.0263 7. Find a mean proportional between 8.76 and 43.5, by logarithms.

Ans. 16.7051 8. Find a third proportional to 12.796 and 3.24718, by logarithms. Ans. .8240216 9. If the interest of 100£, for a year, or 365 days, be 4.5£. what will be the interest of 279,25£. for 274 days.

Ans. 9.433296£.

INVOLUTION.

To find any proposed power of a given number by Logarithms.

Rule. Multiply the Logarithm of the given number by the Index of the proposed power, and the product will be the Logarithm, whose natural number is the power required.

When a negative Index is thus multiplied, its product is negative, but what was carried from the decimal part of the Logarithm must be affirmative; consequently the difference is the index of the product, which difference must be considered of the same kind with the greater, or that which was made the minuend,

EXAMPLES.

1. What is the second power of 3.874 ? Log. of 3.874-0,588160

Index

Power required=15.01=1.176320

2. Required the third power of the number 2.768. Log. of 2.768=0.442166

3. Required the second power of the number.2857. Log. of .2857=-1.455910

Answer=.08162-2.911820

4. Required the third power of the number .7916. Log. of .7916=-1.898506

Index

3

Answer=.4961---1.695518

Hence, 3 times the negative index being 3, and 2 to carry from the decimals, the difference is ī, the index of the product.

5. To find the 4th power of .09163.

Ans. .000070494

To extract any proposed Root of a given number by Logarithms.

RULE.

Find the Logarithm of the given number, and divide it by the Index of the proposed root; the quotient is a Logarithm, whose natural number is the root required.

When the index of the Logarithm to be divided, is negative, and does not exactly contain the divisor without some remainder, increase the index by such a number, as will make it exactly divisible by the index, carrying the units borrowed as so many tens to the left hand place of the decimal, and then divide as in whole numbers.

EXAMPLES.

1. Required the square root of 847. Index 2)2.927883=Log, of 847.

1.463941=Quot.=Log.of 29.103+=Ans.

2. Required the cube root of 847. Index 3)2.927883-Log. of the given number.

0.975961 Quot.-Log. of 9.462=Ans.

3. Required the square root of .093. Index 2)-2.968483-Log, of .093.

[nearly.

-1.484241. Quot.-Log.of.304959-Ans.

4. Required the cube root of 12345. Index 3)4.091491-Log. of 12345.

1.363830=Quot=Log. of 23.116.=Ans.

5. To find the cube root of .00048. Power, or index 3)4.6812412-Log. of the number.

Root.07829735.....2.8937471=Log of the root.

Here, the divisor 3 not being exactly contained in 4, augment it by 2, to make it become 6, in which the divisor is contained just 2 times; and the 2 borrowed being prefixed to the other figures makes 2.6812412, which divided by 3 gives .8937471, therefore 2.8937471 is the Log. of the

root.

6. To find the 4th root of .967845, by Logarithms. Ans. .9918624

7. To find the cube root of 2.987635.

8. To find the cube root of

1

3.14159

Ans. 1.440265

Ans. .6827842

9. To find the value of (.001234).

10. To find the 10th root of 2.

Ans. .00115047

Ans. 1.071773

SECTION IV.

ELEMENTS OF

PLANE GEOMETRY.

DEFINITIONS.

See PLATE I.

1. GEOMETRY is that science wherein we consider the properties of magnitude.

2. A point is that which has no parts, being of itself indivisible; as A.

3. A line has length but no breadth; as AB. figures 1 and 2.

4. The extremities of a line are points, as the extremities of the line AB are the points A and B. figures 1 and 2.

5. A right line is the shortest that can be drawn between any two points, as the line AB. fig. 1. but if it be not the shortest, it is then called a curve line, as AB. fig. 2.

6. A superficies or surface is considered only as having length and breadth, without thickness, as ABCD. fig. 3.

7. The extremities of a superficies are lines.

8. The inclination of two lines meeting one another (provided they do not make one continued line) or the opening between them, is called an angle. Thus in fig. 4, the inclination of the line