be reckoned or called two-hundredths. For example, calling the 9 nine-tenths, or .9, if there be but five spaces between .9 and 1, the first mark will be .92, or ninety-two-hundredths, the second .94, the third .96, the fourth .98, and the fifth onehundred-hundredths, or one. Between the 1 at the middle of the rule, and the 2 following, there are ten principal subdivisions, distinguished by the length of the marks, and consequently each of these must be called one-tenth, or .1; and if there be ten more minor subdivisions between 1 and 2, each of these must be reckoned half of one-tenth, or five-hundredths. The reading, therefore, or reckoning, between 1 and 2, will be 1.1, and 1.2, and 1.3, and so on; or, if we compute the minor subdivisions, the reading will be 1.05, and 1.1, and 1.15, and 1.20, and 1.25, and 1.30, and 1.35, and so on to 2. Beginning again at 1, and calling the 1 ten, then between 10 and 20 each of the principal subdivisions will be equal in value to one unit, and the reading will be 10, 11, 12, 13, 14, and so on to 20. And if between the long marks there be two spaces, each of these must be reckoned one-half, or .5; but if there be five spaces between the principal minor subdivisions, then each of these must be reckoned one-fifth, or .2. Suppose there are fifty spaces or divisions on the rule between 1 and 2, or between 10 and 20, calling the 1 ten and the 2 twenty, then, between 10 and 20, if you reckon the value of each division, you will read 10.2, and 10.4, and 10.6, and 10.8, and 1010, or 11, and 11.2, and 11.4, and 11.6, and 11.8, and so on to 20. Between 20 and 30, if there be ten spaces or divisions, then each division must be reckoned one, but if there be twenty spaces between 20 and 30, each division will be one-half, or .5; and the computation or reading off will be the same between 30 and 40, 40 and 50, 50 and 60, and so on to 80. Between 80 and 90, and 90 and 100, if there be but five divisions, then the value of each will be two. Between 100 and 200, (calling the 1 at the middle of the line 100, and the 2 following 200,) as there are ten long marks, each of these will count ten; and if between these there are but two divisions, each division will count two; but if, instead of two spaces between the long marks, there be five, (which is the more usual number,) then each of these spaces must be called 2; in which case you will read off 102, 104, 106, 108, and so on to 200. Between 200 and 300, if there be but ten spaces, each space will count ten; but if there be twenty divisions each must be reckoned five; and in this case half of one of the divisions would be the half of five, or 2.5. Between 300 and 400, as there are but ten divisions laid down on the rule, each will count ten; and the half of one of the divisions must be reckoned five; and a fourth of one of the spaces would be 2.5; and from 400 to 1000 the computation will be the same, except on those rules which have but five spaces between 800 and 900, and 900 and 1000, when each space must be counted twenty; and the half of one of the divisions must be called ten. Beginning again at the commencement of the scale, and calling the 1 at the beginning one thousand, then the 2 following will be two thousand; and each of the long marks between 1000 and 2000 must be called one hundred and if there be but two spaces between the long marks, then each must be called fifty; but if there be five, then each mark or division will count twenty; and the computation, or reading off, will be the same between 3000 and 4000, and so on. The line of numbers, called the D line, contains the square and cube roots of the numbers expressed on the A, B, or C line; but the manner of reading off is the same on this line as on the others. Under the girt, or D line, is a scale of equal parts, which is useful for many purposes, but more particularly so in drawing plans and mathematical figures, and for laying out work. The lowest line is a scale of inches, one inch being divided into twelve equal parts. The line next above is a scale of equal parts three-fourths of an inch apart; and one of these equal parts is divided into twelve equal parts, one of these minor divisions being of course one-sixteenth of an inch. The line next above is a scale of half-inches, one of the parts being divided into twelve equal parts, and of course expressing one twenty-fourth of an inch; and the upper, or top line, is a scale of fourths of an inch. The lines A and B are used in multiplying, dividing, and stating proportions; and the lines C and D are used in gauging, casting, or finding the contents of squares and solids, and in extracting the square and cube roots. BELCHER'S ENGINEER'S SLIDING RULE. This rule is generally preferred to the carpenter's rule by engineers and machinists. The first three lines (viz. the A, the B, and the C lines) are in all respects the same as the corresponding lines on the carpenter's rule, represented in the cut; but the D line, instead of commencing with 4 on the left end of the scale, commences with 1, and ends with 10 on the right. The principle on which this line is constructed is, nevertheless, the same as that of the corresponding line on the carpenter's rule; and all the gauge points for the latter rule may be used, and with the same facility, by those who possess the former rule; though there are certain methods of solving problems by the engineer's rule, which are not applicable to the carpenter's rule. The D line, as laid down on the engineer's rule, is more simple and easy to be understood than the same line on the carpenter's rule; as it differs in no respect from the other three lines, except that the spaces between the several divisions are twice as great. And since 1 on C stands over 1 on D, (when the slider is in its usual position, or is shut,) the numbers on C are the squares of those standing under them on D; and the numbers on D are the square roots of those directly over them on C. Hence the area of any square, or the side of any square, (containing a given area,) may be found by means of this rule without moving the slider. The principal gauge points, on the line D, will be found between 1 and 2, or between 10 and 20, on the left end of the scale; whilst the same gauge points on the carpenter's rule must be sought for on the right of the 1, near the centre of the rule. Under the line D there is a table of gauge points for square prisms, cylinders, and globes, and likewise gauge points for pumping engines, for the weight of bodies, for regular polygons, for circles, squares, trigons, &c. These gauge points are expressly prepared for this rule, and some of them are not applicable to the carpenter's rule; though the same gauge points may be used on both rules in all cases, when the solution is effected by employing two lines only; and there is no necessity of ever employing more than two lines, at the same time, in solving a problem by either of these rules. To illustrate the method of solving problems by employing three or four lines, and to show the difference between the two rules, we will introduce, in this place, a few examples, which more properly belong to another part of this work. EXAMPLES. 1. Suppose a piece of timber is 9.8 inches square and 20 feet long; how many solid feet does it contain? Ans. 13.3 cubic feet. Set 20, the length, found on B, under 144 on A, and over 9.8, found on D, will be found 13.3 feet on the line C. 2. A cylinder is 30 feet long and 12.73 inches in diameter; how many cubic feet does it contain ? Ans. 26.5 feet. Set 30, the length, found on B, under 183.3, the gauge point on A, and over 12.73, found on D, will be found 26.5 feet on C. 3. A cylinder is 6 inches in diameter and 6 inches in length; how many cubic inches does it contain? Ans. 169 cubic inches. Set 6 on B under 1.273 on A, and over 6 on D, will be found 169 on C. 4. A cylinder is 6 feet long and 20 inches in diameter; required its cubic content. Ans. 14.2 cubic feet. Set 6 on B under 183.3 on A, and over 20 on D, are 14.2 feet on C. 5. What is the content of a cylinder, the content of a globe, and the content of a cone, the diameter of each being 12 inches, and the altitude of the cylinder and cone each 12 inches? Ans. 1356; 904; and 452 cubic inches. Set 12 on B under 1.273 on A, and over 12 on D, are 1356 cubic inches, the content of the cylinder. Set 12 on B under 1.91 on A, and over 12 on D, are 904 inches, the content of the globe. Set 4 on B, (viz. one-third of the altitude of the cone,) under 1.273 on A, and over 12 (the diameter of the base of the cone) on D, are 452 inches on C. These problems may be solved with equal or greater facility by the carpenter's rule, but they cannot be solved in this manner, in consequence of the line D on said rule not being adapted to this method. The gauge points for wine, ale, and imperial gallons, (by the above method,) in a cylinder are 294, 359, and 353 on the line A; and 231,282, and 277.28 for vessels having square bases; and 2150.4 and 2218 for bushels and imperial bushels for square vessels; and 2738.3 and 2824 for cylindric vessels. The weight of bodies may be found in the same manner by means of the gauge points laid down on the face of this rule. A TABLE OF GAUGE POINTS. To find the solid content of shafts or prisms, that are polygonal-sided, by this method. RULE. As the length of the prism on B is to the gauge point on A, so is the length of one of its sides on D to the solid content in inches on C; or set the length upon B to the gauge point upon A, then against the length of one of its sides on D, you have the content in cubic inches on C. |