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EVOLUTION.

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.of29.103+= ans. 2. Required the cube root of 847. Index 3)2.927883=Log: of the given number.

0.975961=Quot. Log.of9.462=ans.

[nearly. 3. Required the square root of .093. Index 2)—2.968483=Log. of .093.

-1.484241=Quot.=Log.of.30-1959= ans.

4. Required the cube root of 12345. Index 3)4.091491=Lg.

of 12345.

1,363830=Quot.=Log. of 23.116.= Ans.

SECTION IV.

ELEMENTS OF

PLANE GEOMETRY.

DEFINITIONS.

Sce 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 4.

3. A line has length but no breadth ; as AB. fi

gures 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 weeting one apother (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 AB to the line BC meeting each other in the point B, or the opening of the two lines BA and BC, is called an angle, as ABC. Note, When an angle is expressed by three let

ters, the middle one is that at the angular point.

9. When the lines that form the angle are right ones, it is then called a right-lined angle, as ÅBC, fig. 4. If one of them be right and the other curved, it is called a mixed-angle, as B. fig. 5. If both of them be curved it is called a curvedlined or spherical angle, as C. fig. 6.

10. If a right line, CD (fig. 7.) fall upon another right line, AB, so as to incline to neither side, but make the angles ADC, CDB on each side equal to each other, then those angles are called right angles, and the line CD a perpendicular.

11. An obtuse angle is that which is wider or greater than a right one, as the angle ADE. fig. 7. and an acute angle is less than a right one, as - EDB. fig. 7.

12. Acute and obtuse angles in general are called oblique angles.

13. If a right line CB. (fig. 8.) be fastened at the end C, and the other end B, be carried quite round, then the space comprehended is called a circle; and the curve line described by the point B, is called the circumference or the periphery of the circle; the fixed point C, is called its centre.

14. The describing line CB. (fig. 8.) is called the semidiameter or radius, so is any line from the centre to the circumference: whence all radii of the same or of equal circles are equal.

15. The diameter of a circle is a right line drawn thro' the centre, and terminating in opposite points of the circumference; and it divides the circle and circumference into two equal parts, called semicircles; and is double the radius, as AB or DE. fig. 8. .

-16. The circumference of every circle is supposed to be divided into 360 equal parts called degrees, and each degree into 60 equal parts called minutes, and each minute into 60 equal parts caled seconds, and these into thirds, fourths, &c. these parts being greater or less as the radius is.

17. A chord is a right line drawn from one end of an arc or arch (that is, any part of the circumference of a circle) to the other; and is the measure of the arc. Thus the right line HG, is the measure of the arc HBG. fig. 8.

18. The segment of a circle is any part there, of, which is cut off by a chord : thus the space which is comprehended between the chord HG and the arc HBG, or that which is comprehended between the said chord HG and the arc IIDAEG are called segments. Whence it is plaing

fig. 8.

1. That any chord will divide the circle into two segments.

2. The less the chord is, the more unequal are the segments.

3. When the chord is greatest it becomes a diameter, and then the segments are equal; and each segment is a semicircle.

19. A sector of a circle is a part thereof less than a semicircle, which is contained between two radii and an arc: thus the space contained between the two radii CH, CB, and the arc HB is a sector. fig. 8.

20. The right sine of an arc, is a perpendicular line let fall from one end thereof, to a diameter drawn to the other end : thus HL is the right sive of the arc HB.

The sines on the same diameter increase till they come to the centre, and so become the radius : hence it is plain that the radius CD is the greatest possible sine, and thence is called the whole sine.

Since the whole sine CD (fig. 8.) must be perpendicular to the diameter (by def. 20.) therefore producing DC'to E the two diameters AB and DE cross one another at right angles, and thus the periphery is divided into four equal parts, as BD, D.1, AE, and, EB; (by def. 10.) and so BD becomes a quadrant or the fourth part of the periphery: therefore the radius DC is always the sine of a quadrant, or of the fourth part of the circle BD.

Sines are said to be of as many degrees as the arc contains parts of 360: so the radius being the sine of a quadrant becomes the sine of 90 deg.ces, or the fourth part of the circle, which is 360 degrees.

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