DESCRIPTION of the SLIDING RULE generally used by Ship- Instructions for delineating the Disposition of all the Timbers composing the Frame; likewise for expanding the Bottom and Topside, by which the Length, Breadth, and Num- ber, of all the Planks may be known, &c. Instructions for delineating the Inboard Works; with Obser- vations on the Inboard Works of Ships in general Instructions for delineating Plans of the Decks, &c. GENERAL EXPLANATION of the METHODS of LAYING-OFF, or of transferring the Lines of the Sheer Draught and other Of the Method of Construction called WHOLE-MOULDING PRACTICAL DIRECTIONS for the ACTUAL BUILDING, 214 228 232 Experimental Method of finding the Tonnage. Mechanical Method of finding the Centre of Gravity in a Ship 253 TABLES for forming the BODIES of several Ships and Vessels in the Royal Navy and Merchant Service; viz. a Seventy-four-gun Ship; a Frigate, of 36 guns; a Merchant Ship, of 330 tons; a Brig of 170 tons: and, a Sloop of 60 tons: together with TABLES of all their principal DIMENSIONS and SCANT- LINGS; and, also, the Dimensions and Scantlings of Boats.. 254 ARITHMETICAL SIGNS AND CONTRACTIONS *The Reader is particularly requested to attend to the Errata, &c. at the End of the Volume. INTRODUCTION. SECTION I. OF ARITHMETIC. THE first rudiments of ship-building have their foundation in the princi ples of arithmetic and practical geometry; it is therefore proper that this treatise should commence with an explanation of these sciences. It is, however, to be presumed, that every young artist, before he proceeds to the study of the art of ship-building, will, at least, have acquired a competent knowledge of the fundamental rules of common arithmetic; hence it would be superfluous to explain them here: we shall, consequently proceed, in the first instance, briefly to explain the more complex rules, particularly proportion, vulgar and decimal fractions, the extraction of roots, &c. commencing with, PROPORTION IN GENERAL. By proportion is meant that analogy or relation which two or more numbers have to each other, with respect to their comparative or relative values; and by which, when compounded together, those values may be ascertained. In a proportional statement of two numbers, the number first written is called the antecedent; the latter the consequent; and the antecedent is compared with the consequent, to form a comparison. Arithmetical progression consists in the difference between the antecedent and consequent being always the same, through a series of numbers. Geometrical progression considers the quotient of the two numbers, which is called the geometrical ratio. Thus, 4, 3, 2, 1, are in arithmetical progression, the constant difference being 1: but 4, 8, 16, 32, &c. are in geometrical progression, because the quotient produced by the division of any one of these numbers by another (provided the divisor be less) is 2; the number 2, therefore, is the geometrical ratio. When all the greater or inferior terms of two or more couplets of numbers are similarly stated, that is, when all of one sort are taken as antecedents, and the other as consequents; if the ratio or difference of each couplet be the same, the couplets are said to be in proportion; and their terms are called proportionals. Thus in the two couplets 4, 6, and 8, 10, if they are placed thus, 4, 6, 8, 10, or thus, 6, 4, 10, 8, they are arithmetical proportionals-and if the two couplets 4, 8, 16, 32, be taken thus, 4, 8, 16, 32, or thus, 8, 4, 32, 16—they are geometrical proportionals. Proportion may be divided into continued and discontinued. If the difference or ratio of the consequent of one couplet, and the antecedent of the succeeding one be the same as the common difference or ratio of the couplets, the proportion is continued; if not, it is discontinued. Thus, 4, 6, 8, 10, form a continued arithmetical progression, because the common difference is 2, and 6-8=2. Also the couplets 4, 8, 16, 32, &c. form a continued geometrical progression, because =2, 16=2, &c. the ratio being 2. But 6, 4, 10, 8, are in discontinued arithmetical proportion, &c. 8, 4, 32, 16, are in discontinued geometrical proportion, because the ratio in both cases is two, but 10-4-6, and 32÷4=8. When the terms gradually increase, the series is called ascending; when they decrease, descending. To denote geometrical proportion, the couplets are separated by a double colon::--and a colon is written between the terms of each couplet. Thus, 4: 8:: 16: 32; that is to say, as 4 is to 8, so is 16 to 32. OF ARITHMETICAL PROGRESSION. ARITHMETICAL progression is the augmentation or diminution of any series of numbers, by the addition or subtraction of an equal difference. Thus, 2, 4, 6, 8, 10, are numbers in arithmetical progression increasing by 2; and 100, 96, 92, 88, 84, 80, 76, 72, &c. are in arithmetical progression, decreasing by 4. PROPOSITION I. To find the sum of the series, multiply the sum of the extremes by half the number of terms. The product will be the answer. EXAMPLE 1. The sum of a number of terms, increasing by 3, the extremes being 1 and 25 ? Here it is evident, that as the difference between 1 and 25 is 24, and the common difference 3, as 24-3=8, the number of terms must be equal to 8+the 1st term (1); i. e. the number of terms 9. Sum of 1+25=26. 26×4117 the Answer. EXAMPLE 2. What is the sum of the number of times that a clock strikes between the hours of 2 and 12. The first term=3; the last term=12; the common difference 1-the number of terms=10. 3+12=15, 15 x 5=75 the Answer. PROPOSITION II. Given one of the extremes, the common difference, and the number of terms, to find the other extreme. RULE. Multiply the common difference by one less than the number of terms; then add the product to the first term, to find the greatest extreme; or deduct it from the largest extreme to find the least. The largest extreme is required of a number whose other extreme is 5, the number of terms 10, and the common difference 2 ? 2x9=18. 18+5=23 the Answer. The smallest extreme is required. of a number whose greatest term is 103; the common difference 4, the number of terms 25. 4× 24-96. 103-96-7 the Answer. PROPOSITION III. Given the extremes and common difference, to find the number of terms. .Deduct the lesser extreme from the greater, divide by the common difference, add 1 to the quotient. The sum is the Answer. Extremes 48 and 152. Common difference 4. Required number of terms? 152-48104. 104 4 26. 26+1=27 the Answer. PROPOSITION IV. Given the extremes and number of terms to find the common difference. Deduct the lesser extreme from the greater. Divide the difference by the number of terms -1. The quotient is the common difference. EXAMPLE. The 2 extremes 589 and 1093; the number of terms=127. Required the common difference? 1093-589-504. 504 126=4 the common difference. OF GEOMETRICAL PROGRESSION. A GEOMETRICAL PROGRESSION is a series of numbers, increasing or de creasing by a given proportion, so that each term divided by its preceding one will produce the same quotient that is given by the division of a number, to which the afore-mentioned term is the divisor, and the next adjacent increasing or decreasing term the dividend. Thus, 2:4:8:16:: 32:64:: 128, &c. 8-4 produces the same quotient or common ratio, as 16-8, &c. 32:16:8:4, &c. 168 gives 2; as also 8 by 4, &c. PROPOSITION I. To find the sum of the series. Divide the difference of the extremes by the ratio less 1. Add to the quotient the greater extreme. The sum is the sum of the series. EXAMPLE. The lesser extreme being 2, the larger 256, the ratio 2, required the sum of the series? 2:4;:8:16:: 32: 64: : 128: 256: ; the number of terms 8. |