EXAMPLES. 1st. If a clock gain 14 seconds in 5 days 18 hours, how much will it gain in 17 days 15 hours? 5.75 days : Log. = 0.759668 Answer =42". 91 = 1.632589 Or thus; 5.75 days: Arith. Co. Log. 17.625:: 14 seconds: = 9.240332 Answer = 42′′.91 = 1.632589 2d. Find a fourth proportional to 98.45, 1.969, and 347.2. 98.45 : Log. = €1.993216 3d. What number will have the same proportion to .8538 as .3275 has to .0131 ? 4th. Required a third proportional number to 9.642 and 4.821. 9.642 : Log. 0.984167 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. Ans. 16.7051. Ans. .8240216. 7. Find a mean proportional between 8.76 and 43.5, by logarithms. 8. Find a third proportional to 12.796 and 3.24718, by logarithms. 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. £9.433294. 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. Power required = 15.01 = 1.176320 2. Required the third power of the number 2.768. Log. of 2.768: 0.442166 Index = 3 = 1.326498 3. Required the second power of the number .2857. 4. Required the third power of the number .7916. Hence, 3 times the negative index being-3, and 2 to carry from the decimals, the difference is -1, the index of the product. 5. To find the 4th power of .09163. EVOLUTION. Ans. .000070494. Ans. 36.72203. Ans. 28.97575. Ans. 1.909864. Ans. 5.148888. 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. 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 fourth root of .967845, by logarithms. Ans. .9918624. 7. To find the cube root of 2.987635. Ans. 1.440265. 8. To find the cube root of 3.14159 Ans. .6827842. 9. To find the value of (.001234). Ans. .0115047. 10. To find the tenth root of 2. 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 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 letters, 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 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 curved-lined 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 through 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 compre |