For the extraction of the cube root, see Tables of Square and Cube Roots. ARITHMETICAL PROGRESSION. This subject is often referred to in elementary books on mechanical science; and for this reason, we shall draw the attention of the reader for a little while to it. ARITHMETICAL PROGRESSION is a series of numbers which succeed each other regularly, increasing or diminishing by a constant number or common difference: As 1, 3, 5, 7, 9, &c. increasing series. 15, 12, 9, 6, 3, &c. I decreasing series. The numbers which form the series are called terms. The first and the last term are called the extremes, and the others are called the means. In arithmetical progression, there are five things to be considered, viz.: 1, The first term. 2, The last term. 3, The common difference. 4, The number of terms. 5, The sum of all the terms. These quantities are so related to each other, that when any three of them are given, the remaining two can be found. Given the First Term, the Common Difference, and the Number of Terms, to find the Last Term. Rule.-Multiply the number of terms, less one, by the common difference, and to the product add the first term. Example-What is the 20th term of the arithmetical progression, whose first term is one, the common difference? 20-1=19 and 19×1=91; and 91+1=101. Ans. Given the Number of Terms and the Extremes, to find the Common Difference. Rule. Divide the difference of the extremes by one less than the number of terms. Example. The extremes are 3 and 29, and the number of terms 14, required the common difference. 29-3=26; and 26÷13=2. Ans. Given the Common Difference and the Extremes, to find the Number of Terms. Rule. Divide the difference of the extremes by the common dif ference, and to the quotient add one. Example. The first term of an arithmetical progression is 11, the last termi 88, and the common difference 7. What is the number of terms? 88-11=77; and 77÷7=11; 11+1=12. Ans. Given the Extremes and the Number of Terms, to find the Sum of all the Terms Rule.-Multiply half the sum of the extremes by the number of terms. Example.-How many times does the hammer of a clock strike in 12 hours? 1+12 13 the sum of the extremes. Then 12×(13÷2)=78. Ans. The following formula will be found much shorter, and very convenient. Let a be the first term; d the common difference; n the number of terms; 7 the nth term; and s the sum of nth terms. Thus: la+d(n−1); s=(a+b). GEOMETRICAL PROGRESSION. Geometrical Progression is any series of numbers which succeed each other regularly by a constant multiplier, or decrease by a constant divisor. The constant multiplier or divisor is called the ratio. As, 1, 3, 9, 27, 81, &c., is an ascending geometrical progression, whose ratio is 3; and 15, 71, 33, is a descending geometrical progression, whose ratio is. In geometrical progression, as in arithmetical progression, when any three of the following parts are given, the remaining two can be found, viz. The first term, the last term, the number of terms, the ratio, and the sum of all the terms. Given the Ratio, the Number of Terms, and the First Term, to find the Last Term. Rule. 1. Write some of the leading powers of the ratio, and over them place their several indices, beginning with a cipher. 2. Add together the most convenient indices to make an index less by 1 than the number of terms sought. 3. Multiply together the powers or terms of the series standing under those indices; and their product, multiplied by the first term, will be the answer sought. Example.-The first term of a geometrical series is 4, the ratio 3 and the number of terms 11. What is the last term? 1, 2, 3, 4, 5. 3, 9, 27, 81, 243. The indices 5+3+2=10. Then, 9×27×243-59049, which, multiplied by the first term 4=236196, the last term required. Given the Ratio, the First Term, and the Number of Terms, to find the Sum of all the Terms. Rule.-Raise the ratio to a power whose index is equal to the number of terms, and from this power subtract 1; divide the remainder by the ratio, less 1, and multiply the quotient by the first term for the answer. Example. The first term of a geometrical progression is 2, the ratio 3, and the number of terms 6. What is the sum of all the terms? 36729 and -1=728 728-(3-1)x2=728. Ans. PERMUTATION. Permutation is the method for ascertaining how many different ways any given number of persons or things may be varied in their positions. Rule.-Multiply all the terms of the natural series continually together, and the last product will be the number of changes required Example-How many changes may be made by 8 scholars in seating themselves differently at their recitations? 1 2 3 4 5 6 7 8, and 1×2×3×4 × 5 × 6 × 7×8=40320 times. Ans. MENSURATION. MENSURATION OF SUPERFICIES. SECTION I. MENSURATION is that branch of mathematics by which we ascertain the contents or superficial areas, and the extension, solidities, and capacities of bodies. The area, or superficial contents of any figure, is the measure of its surface, or the space contained within the bounds of that surface, without any regard to thickness. In calculating the area, or the contents of any plane figure, some particular portion of surface is fixed upon as the measuring unit, with which the figure is to be compared. This is commonly a square, the side of which is the unit of length, being an inch, or a foot, or a yard, or any other fixed quantity, according to the measure peculiar to different artists; and the area or contents of any figure is computed by the number of those squares contained in that figure. For the same reason, determining the quantity of surface in a figure is called squaring it; that is, determining the square or number of squares to which it is equal. In order to form correct estimates of the extent of surfaces and solids, various rules have been adopted, most of which, the most valuable and useful in practice, will be found accompanying their respective problems in the following treatise, and with which the mechanic may speedily perform all the calculations that ordinarily occur in the practical details of his business. . DEFINITIONS. The following definitions, which are similar in substance to those found in Euclid, are here inserted for the convenience of reference. I. Four-sided figures are variously named, according to their relative position and length of their sides. 1. A line is length, without breadth or thickness. 2. Parallel lines are always at the same perpendicular distance and they never meet, though ever so far produced. 3 An angle is the inclination or opening of two lines, having dif ferent directions, and meeting in a point. 4. A parallelogram has its opposite sides parallel and equal. 5. A rectangle, or right parallelogram, has its opposite sides equal, and all its angles right angles. 6. A square is a figure whose sides are of equal length, and all its angles right angles. 7. A rhomboid has its opposite sides equal, and its angles oblique 8. A rhombus is an equilateral rhomboid, having all its sides equal, but its angles oblique. 9. A trapezoid is a quadrilateral figure, having only two of its sides parallel. 10. A trapezium is an irregular figure, of four unequal sides and angles. II. When figures have more than four sides, they are classed under the head of Polygons. These again are either regular or irregular, according as their sides and angles are equal or unequal, and they are named from their number of sides or angles. Thus, a regular polygon has all its sides and angles equal. III. A figure of three sides and angles is called a triangle, and receives particular denominations from the relations of its sides and angles. 1. An equilateral triangle is that whose three sides are equal. 2 The height of a triangle is the length of a perpendicular drawn from one of the angles to the opposite side. 3. An isosceles triangle is that which has only two sides equal. 4 The height of a four-sided figure is the perpendicular distance between two of its parallel sides. To find the Area of a Four-sided Figure, whether it be a parallelogram, square, rhombus, or rhomboid. Rule.-Multiply the length by the breadth or perpendicular height, and the product will be the area. Example.-What is the area of a parallelogram, a b c d, whose length, cd, is 12 feet 3 inches, and whose breadth, a c, is 8 feet 6 inches? |