into lineal feet produce solid feet; superficial inches multiplied into lineal inches produce solid inches, &c. 2. Superficial feet into lineal inches produce parallelopipeds, whose base is 1 square foot, and their height 1 inch; which divided by 12 quote solid feet, and the remainder multiplied by 144, produces solid Inches. 3. Superficial feet into lineal lines produce parallelopipeds, whose base is 1 square foot, and their height 1 line; which divided by 144 quote solid feet; and the remaindar multiplied by 12 produces solid inches. 4. Superficial inches into lineal lines, produce parallelopipeds, whose base is 1 square inch, and their height one line; which divi ded by 12 quote solid inches; and the remainder multiplied by 144 produces solid lines. 5. Lineal feet into superficial inches produce parallelopipeds, whose base is one square inch, and their height 1 foot; which divided by 144 quote solid feet; and the remainder multiplied by 12 produces solid inches. 6. Lineal feet into superficial lines produce parallelopipeds, whose base is 1 square line, and their height 1 foot; which divided by 12 quote solid inches; and the remainder multiplied by 144, produces solid lines. 7. Lineal inches into superficial lines produce parallelopipeds, whose base is 1 square line, and their height 1 inch; which divided by 144 quote solid inches; and the remainder multiplied by 12 produces solid lines. EXAMPLE. How many solid feet in a piece of timber whose length is 18 feet 6 inches, breadth 2 feet 4 inches, and thickness 2 feet 3 inches ? Here I first multiply 18 feet 6 inches by 2 feet 4 inches, and the product is 43F. 2 inches, superficial, which I next multiply by 2 feet 3 inches, lineal produce 97 feet and 216 inches. 97 1 6 solid content; or 97 feet 216 inches; 1 i. 6 equal to 216 inches. Or thus : p. being 861 $2 10 1296 5761 197 216 EXPLANATION. Here I multiply 18 feet 6 inches, by 2 feet 4 inches, as formerly taught, and the product is 43 feet 24 inches superficial, which I next multiply by 2 feet 3 inches lineal; thus 43 superficial feet into 2 lineal feet, produce 86 feet solid; and 24 superficial inches into 3 lineal inches, produce 72 solid inches, by rule 1. Then 43 superficial feet, into 3 lineal inches, produce 129 parallelopipeds, whose base is 1 square foot, and their height 1 inch; which divided by 12 quotes 10 solid feet; and the remainder 9, multiplied by 144, produce 1296 solid inches, by rule 2d. Again, 2 lineal feet into 24 superficial inches produce 48; which being less than 144, I esteem a remainder, and multiplying it by 12, I have a product of 576 solid inches, by rule 5th. Because the sum of the inches exceeds 1728, I carry 1 from them to the feet, and the overplus, 216, I set down. EXAMPLES. 1. How many solid feet in a polished stone, that is 8 feet 9 inches 5 parts long, 7 feet 3 inches broad, and 3 feet 5 parts thick. F. I. P. 3X36=108, and 12)108(9 in. 2. What is the difference in time, of the sun's coming to the meridian of two places, whose difference of longitude is 31° 27′ 30", the sun passing through 15o in an hour? Ans. 2h. 6' 30" RULE OF THREE DIRECT. The Rule of Three, called also, on account of its excellence, the Golden Rule, from certain numbers given, finds another; and is divied into Simple and Compound, or into Single and Double. SIMPLE OR SINGLE RULE OF THREE DIRECT. The Single Rule of Three teaches from three numbers given to find a fourth, which bears the same proportion to the third as the second does to the first. The nature and properties of proportional numbers, may be understood sufficiently for our purpose, from the following observations. In comparing any two numbers, with respect to the proportion which the one bears to the other, the first number, or that which bears proportion, is called the antecedent; and the other, to which it bears proportion, is called the consequent ; and the quantity of the proportion or ratio is estimated by the quotient arising from dividing the antecedent by the consequent. Thus, the ratio or proportion between 6 and 3 is the quotient arising from dividing the antecedent, 6, by the consequent, viz. 2; and the ratio or proportion between 1 and 2, is the quotient arising from the division of the antecedent 1, by the consequent 2, viz. . Four numbers are said to be proportional when the ratio of the first to the second is the same as that of the third to the fourth; and the proportional numbers are usually distinguished from one another, as in the following examples. 4: 2:16:8 6 : 9:: 12: 18 Proportional numbers, or numbers in proportion, are usually denominated terms; of which the first and last are called extremes, and the intermediate ones get the name of means, or middle terms.. If four numbers are proportional, they will also be ivnersely proportional; that is, the first consequent will be to its own antecedent as the second consequent is to its antecedent; or the fourth term will be to the third as the second is to the first. Thus, if 6 : 3 :: 10:5, then by inversion, 3 : 6 :: 5 : 10, or 5: 10 :: 36. See Euelid, the V. of his Elements of Geometry. By either of these kinds of inversion may any question in the rule of three be proved. If four numbers be proportional. they will also be alternately proportional; that is, the first antecedent will be to the second antecedent as the first consequent is to the second consequent; or the first term will be to the third term as the second term is to the fourth. Thus, if 8 : 4 :: 24: 12. Then, by alternation, 8: 24 :: 4 : See Euclid, V. 16. 12 But the celebrated property of four proportional numbers is, that the product of the extremes is equal to the product of the means. Thus, if 23: 69, then 2X9=3×6=18. Eu. VI. 16. Hence we have an easy method of finding a fourth proportional to three numbers given, viz. Multiply the middle number by the last, and divide the product by the first; the quotient gives the fourth proportional. EXAMPLE. Given 6, 5, and 36, to find a fourth proportional; put X, equal to the fourth proportional, then 6: 5 :: 36: X, and 5X 36 =180=6XX; wherefore, dividing the product 180, by the factor 6, the quotient gives the other factor X, namely 30, the fourth propor tional sought. Every question in the rule of three may be divided into two parts, viz. a supposition and a demand; and of the three given numbers two are always found in the supposition, and only one in the demand. EXAMPLE. If 4 yards cost 12 dollars, what will 6 yards cost at that rate? In this question the supposition is, if 4 yards cost 12 dollars; and the two terms contained in it are 4 yards and 12 dollars; the demand lies in these word, what will 6 yards cost? and the only term found in it is 6 yards. The supposition and demand being thus distinguished, proceed to state the question, or to put the terms in due order for operation, as the following rules direct. RULES. 1. Place that term of the supposition which is of the same kind with the number sought in the middle. The two remaining terms are extremes, and always of the same kind. 2. Consider, from the nature of the question, whether the answer must be greater or less than the middle term; and if the answer must be greater, the least extreme is the divisor; but if the answer must be less than the middle term, the greatest extreme is the divisor. 3. Place the divisor on the left hand, and the other extreme on the right; then multiply the second and third terms together, and divide their product by the first, and the quotient gives the answer; which is always of the same name with the middle term. When the divisor happens to be the extreme found in the supposition, the proportion is called direct but when the divisor happens to be the extreme in the demand, the proportion is inverse. The three above rules are so framed as to preclude the distinction of direct and inverse, or render it needless,; but yet the direct questions being plainer in their own nature, and more easily comprehended by a learner, I shall in the first place exemplify the rules by a few questions of the direct kind, and shall afterwards adduce a few of such as are inverse. A general rule to know whether a question belongs to the rule of three direct or inverse. If more requires more, or less requires less, the question belongs to the rule of three direct; but if more require less, or less require more, it belongs to the rule of three inverse. NOTE. More requiring more is when the third term is greater than the first, and requires the fourth term to be greater than the second. And less requiring less is when the third term is less than the first, and requires the fourth term to be less than the second. Also, more requiring less, is when the third term is greater than the first, and requires the fourth term to be less than the second. And less requiring more is when the third term is less than the first, and requires the fourth term to be greater than the second. SIMPLE RULE OF THREE DIRECT. EXAMPLES. 1. If 4 yards of cloth cost 12 dollars, what will 6 yards cost at that rate? The supposition and demand of this question have already been distinguished, and the two terms in the former are 4 yards and 12 dollars, and the only term in the latter is 6 yards. The number sought is the price of 6 yards, and the sum in the supposition of the same kind, is the price of 4 yards, viz. 12 dollars, which I place in the middle, as directed in rule 1, and the two remaining terms are extremes, and of the same kind, viz. both lengths. OPERATION. yds. $ yds. 4)72 Ans. 18 dollars. EXPLANATION. It is easy to perceive that the answer must be greater than the middle term; for 6 yards will cost more than 4 yards; therefore the least extreme, viz. 4 yards, is the divisor, according to rule 2. Wherefore I place the divisor, 4 yards, on the left hand, and the other extreme 6 yards on the right; and multiplying the second and third terms together, I divide their product by the first term, and the quotient, 18, is the answer, and of the same name with the middle term, viz. dollars, according to rule 3. And because the divisor is the extreme found in the supposition, the proportion is direct. NOTE. The common method of proving the rule of three is by inverting the question; but I recommend the following, viz. If 4 numbers be proportional, the product of the extremes will be equal to the product of the means. Thus, |