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 2 Power required=15.01 =1,176320 2. Required the third power of the number 2.768. Log. of 2.768=0.442166 3 Answer=21.21=1.326498 3. Required the second power of the number.2857. Log. of.2857=-1.455910 2 Answer=,08162=-2.911820 4. Required the third power of the number .7916. Log. of.7916=-1.898506 3 Answer=,4961=-1.695518 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.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. (nearly. 3. Required the square root of .093. Index 2)—2.968483=Log. of.093. -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 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. 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 meeting one another (provided they do not make one continued line) or the opening between them, is called an ters, the middle one is that at the angular 9. When the lines that form the angle are right ones, it is then called a right-lined angle, as ABC, 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 called 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 thereof, 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 HDAEG are called segments. Whence it is plain, 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. |