Extracts from A System of
Practical Mathematics
|
||
The Description and Use of the Sliding-Rule(pages 169-172) Soon after the Invention of Logarithms, Mr. Edmund Gunter contrived the laying of them on a Ruler or Scale, which from him was called Gunter’s Scale; on which, with the Help of a Pair of Compasses, any Question relating to Proportion may be speedily wrought: Since that Mr. Oughtred contrived to make two Scales slide one by the other, which performs the Work without Compasses, and is called the Sliding-Rule: Of these there are several Sorts; but, as that usually called Coggeshall’s is the most common and best known, I shall give the Description and Use of that. This is usually put on one of the Sides of a Carpenter’s Joint-Rule, by a Piece made to slide in a Groove: This sliding Piece, as also the Line above it, is divided and numbered 1, 2, 3, &c. to 9, and again from 1 in the Middle, to 10 at the End: These two are called the Lines of Numbers; each of those Divisions betwixt the Figures is divided into ten Parts; and those Divisions, from 1 at the Beginning and Middle to the 2, are divided into five Parts, and, from each 2, to each 3, into Halves; the divisions being afterwards too small to be divided. The Sliding-Piece has two of those Lines on it; one on the upper Edge, to answer that on the Rule; another on the under Edge, to answer the Girt-Line on the Rule, below the Slider: These four Lines are marked A, B, C, D. The Girt-Line is a broken Line of Numbers, contrived for measuring Timber at one Operation: It is numbered from 4 at the Beginning, to 5, 6, 7, 8, 9, 10, 20, 30, 40, at the End; each Division, from the Beginning to 8, divided into ten, and those again halved; but, from 8 to 9, and from 9 to 10, are only divided into ten Parts; then, from 10 to the End, each of the numbered Divisions is divided, first into ten Parts, and each of those Tenths into four smaller Divisions. To find a given Number on the Line of Numbers. In the first Part of the Line, to the left Hand, if the Numbers 1, 2, 3, &c. be accounted as Units, then the larger Divisions are, each, tenth Parts of an Unit, and the smaller Divisions are parts of those Tenths: The 1 in the Middle of the Line stands for 10, and the other Figures 2, 3, &c. to the End, stand for 20, 30, 40, &c. to the 10 at the End, which stands for 100. In this Case, therefore, all Numbers under 10 are to be look’d for in the first Part of the Line, and Numbers from 10 to 100 in the last Half: Thus, if it be required to find 12 on the Line, the 1 in the Middle standing for 10, count from thence two of the larger Divisions, and you’ll find this mark’d 12: If 37 were requir’d, from 3 on the last Part of the Line count seven of the Divisions; and so of any other Number under 100. If it were required to find 2.48, from 2 in the first Part of the Line, count four of the larger Divisions, this is 2.4; then, as each of these Divisions are here divided into Half, somewhat more than half Way betwixt this smaller Division and the Fifth of the larger, is the place of .8 of the larger Divisions, which is .08 of the number’d Divisions, and is the Place of 2.48, the given Number. The same Way may other decimal Parts be found, either in the first or last Part of the Line. Of the Use of the Line of Numbers. To multiply one Number by another, set 1 on the Line B to the Number given for the Multiplier in the Line A; then will the Number given for the Multiplicand in the Line B stand against the Product in the Line A. Example: To multiply 8 by 7, set 1 in the Line B against 7 in the Line A; then will 8 in the Line B stand against 56, the Product, in the Line A; and while, the Rule is in this Position, you have the Products of all the Numbers on the Line B, from 1, to so far as the Line A reaches, multiplied by 7, and the Products in the Line A: Thus 2, 3, 4, 5, &c. on the Line B, stand against 14, 21, 28, 35, &c. in the Line A. The same is the Case of any mixed Number found on the Line B; thus 1.6 stands against 11.2 in the Line A. To divide one Number by another, set the Divisor on the Line B against 1 in the Line A; then against the Dividend in the Line B, you have the Quotient in the Line A. Example: To divide 96 by 6, set 1 in the Line A against 6 on the Line B; then will 96 on the Line B stand against 16, the quotient, in the Line A; and while the Rule is in this Position, you have the Quotient on the Line A of all the Numbers on the Line B, divided by 6. For the Measuring of Timber, the Girt-Line Mark’d D is particularly design’d; for the Use of which take the following Examples. Example 1. A piece of Timber 10½ Inches square, 44 feet long, to find the Content. Set the 12 on the Line D to 44 on the Line C; then will 10½ on the Line D stand against 33.68 Feet, the Content; which is 33 Feet, 8 Inches. Example 2. A Piece of Timber, 22 Inches square, 34 Feet long, to find the Content in Feet. Set 12 on the Girt-Line D to 34, the Length of the Piece of Timber in Feet, on the Line C; then will 22 on the Line D, stand against the Content in Feet on the Line C: But here it is to be observed, that 22 on the Line D, will reach beyond the Line C, the Content being more than 100 Feet: Therefore, in this Case, the 12 in the Girt-Line D must be set to 34 in the former Part of the Line C, accounting the Figures as Tens, the same as in the latter Part; then the 1 in the Middle is 100, the 2, 3, 4, &c. following, 200, 300, 400, &c. to the 10 at the End, which is now 1000. Hence the larger Divisions, which before were Units, are in this Case Tens, and the smaller divisions increase their Value in the same Proportion. Having set the 12 in the Line D against 34 in the first Part of the Line C, then will 22, the Inches square in the Line D, stand against 114, and somewhat more; that is, it will stand a little beyond the second small Division, betwixt the first and the second of the large Divisions, answering to 114.277 Feet, or 114 Feet, 3 Inches. By the Method, used in this Example, of making the Numbers in the first Part of the Line stand for Tens (or Hundreds, if necessary) the use of the Sliding-Rule may be extended to larger Numbers; always remembering, that all the Numbers on the last Part of the Scale will be of the same Denomination with the 1 in the Middle; and, though in large Numbers we cannot come to the Degree of Exactness necessary in some things, yet it may be of great Use to correct a Mistake. This page was last updated on 30 July 2000. |