The converse of this (that is, to reduce English miles into nautical miles,) must appear obvious. Remark 2.-This Table was computed by the following rule; viz., To the earth's diameter in feet, add the height of the eye above the level of the sea, and multiply the sum by that height; then, the square root of the product being divided by 6080 (the number of feet in a nautical mile), will give the distance at which an object may be seen in the visible horizon, independent of terrestrial refraction. This rule may be adapted to logarithms, as thus: Let the earth's diameter in feet be augmented by the height of the eye; then, to the log. thereof add the log. of the height of the eye; from half the sum of these two logs. subtract the constant log. 3. 783904,* and the remainder will be the log. of the distance in nautical miles, which is to be increased by a twelfth part, of itself, on account of the terrestrial refraction. Example. At what distance can an object be seen, in the visible horizon, by an observer whose eye is elevated 290 feet above the level of the sea? Distance uncorrected by refraction 18.11=Log. = 1.257907 Note. For the principles of this rule, see how the distance of the visible horizon, expressed by the line O T, is determined in page 5. This is the log. of 6080, the number of feet in a nautical mile. TABLE LVI. To reduce the French Centesimal Division of the Circle into the English Sexagesimal Division; or, to reduce French Degrees, &c., into English Degrees, &c., and conversely. This Table is intended to facilitate the reduction of French degrees of the circle into English degrees, and conversely. The Table is divided into two parts: the first or upper part exhibits the number of English degrees and parts of a degree contained in any given number of French degrees and parts of a degree; and the second or lower part exhibits the number of French degrees, &c., contained in any given number of English degrees, &c. Note. In the general use of this Table, when any given number of French degrees exceeds the limits of the first part, take out for 100 degrees first, and then for as many more as will make up the given number; and, when any given number of English degrees exceeds the limits of the second part, take out for 90 degrees first, and then for as many more as will make up the given number. Example 1. If the distance between the moon and a fixed star, according to the French division of the circle, be 128:93:96", required the distance agreeably to the English division of the circle? Distance reduced to English degs., as required 116: 2:44′′.30 Example 2. If the distance between the moon and sun, according to the English division of the circle, be 116:53.47", required the distance agreeably to the French division of the circle? Distance reduced to French degs, as required = 129:88:48".76 Remark 1.-This Table was computed in conformity with the following considerations and principles; viz., The French writers on trigonometry have recently adopted the centesimal division of the circle, as originally proposed by our excellent countryman Mr. Henry Briggs, about the year 1600. In this division, the circle is divided into 400 equal parts or degrees, and the quadrant into 100 equal parts or degrees; each degree being divided into 100 equal parts or minutes, and each minute into 100 equal parts or seconds: these degrees, &c. &c., are written in the usual manner and with the customary signs, as thus; 128:93.96". Hence, the French degree is evidently less than the English, in the ratio of 100 to 90; a French minute is less than an English minute, in the ratio of 100 x 100' to 90 × 60; and a French second is less than an English second, in the ratio of 100: × 100 × 100% to 90 × 60 × 60% : now, the converse of this being obvious, we have the following general rule for converting French degrees into English, and the contrary. As 100, the number of degrees in the French quadrant, is to 90, the number of degrees in the English quadrant; so is any given number of French degrees to the corresponding number of English degrees. As 10000, the number of minutes in the French quadrant, is to 5400, the number of minutes in the English quadrant; so is any given number of French minutes to the corresponding number of English minutes. And, As 1000000, the number of seconds in the French quadrant, is to 324000, the number of seconds in the English quadrant; so is any given number of French seconds to the corresponding number of English seconds. English degrees, minutes, and seconds, are reduced into French by a converse proportion; viz., As 90, is to 100; so is any given number of English degrees to the corresponding number of French degrees. As 5400, is to 10000; so is any given number of English minutes to the corresponding number of French minutes. And, As 324000, is to 1000000; so is any given number of English seconds to the corresponding number of French seconds. Remark 2.-French degrees and parts of a degree may be turned into English, independently of the Table, by the following rule; viz., Let the French degrees be esteemed as a whole number, to which annex the minutes and seconds as decimals; then one-tenth of this mixed number, deducted from itself, will give the corresponding English degrees, &c. Example. The latitude of Paris, according to the French division of the quadrant, is 34:26:36 north; required the latitude agreeably to the English division of the quadrant? Given latitude 54°26'36"=54°.2636 = Hence, the latitude of Paris, reduced to the English division of the quadrant, is 48:50:14" north. Remark 3.-English degrees and parts of a degree may be turned into French, independently of the Table, as thus : Reduce the English minutes and seconds to the decimal of a degree, and annex it to the given degrees; then one-ninth of this mixed number, being added to itself, will give the corresponding French degrees, &c. Example. The latitude of the Royal Observatory at Greenwich is 51:28:40" north, agreeably to the English division of the quadrant; required the latitude according to the French division of the quadrant ? Given latitude 51:28:40 51°.4777777, &c. = Hence, the latitude of Greenwich Observatory, according to the French division of the quadrant, is 57:19:75". 307 N. TABLE LVII. A general Table for Gauging, or finding the Content of all Circularheaded Casks. Although this Table may not directly affect the interest of the mariner; yet, since it cannot fail of being exceedingly useful to officers in charge of His Majesty's victualling stores (such as Pursers of the Royal Navy, Lieutenants commanding gun-brigs, &c. &c.), it has therefore been deemed advisable to give it a place in this work, particularly since it may be found interesting to those whom it immediately concerns. This Table is divided into two parts: the first part consists of five compartments, and each compartment of three columns; the first of which contains the quotient of the head diameter of a cask divided by the bung diameter; the second the corresponding log. adapted to ale gallons; and the third the log. for wine gallons. The second part of the Table contains the bung diameter and its corresponding logarithm. The use of this Table will be exemplified in the following PROBLEM. Given the Dimensions of a Cask, to find its Contents in Ale and Wine Gallons. RULE. Divide the head diameter by the bung diameter to two places of decimals in the quotient; then add together the log. for ale or wine gallons, corresponding to this quotient, in the first part of the Table; the log. corresponding to the bung diameter, in the second part of the Table, and the common log. of the length of the cask; the sum of these three logs., rejecting 10 in the index, will be the log. of the true content of the cask, in ale or wine gallons, according as the content may be required. Example. Let the bung diameter of a cask be 25 inches, the head diameter 19.5 inches, and its length 31 inches; required the contents in ale and wine. gallons? 25)19.50(.78, quotient of the head diameter divided by the bung diameter. 175 |