SIGNS AND TABLES USED IN THIS WORK. 1. ARITHMETICAL SIGNS. + Plus, or more. Sign of addition, as 8+ 4 = 12; 8 and 4 are 12. = Equality, or equal to. As 3 feet added to 3 feet Minus, or less. Sign of subtraction; as 8 -4 4 remain. = 6 feet. = 4; 4 from 8 and X Into, or by. Sign of multiplication, as 4 x 2: = 8; 4 multiplied by 2 is 8. Divided by. 25 Sign of division; as 25 ÷ 5 = 5; 5 is contained in 25, 5 times, or, = 5; 5 10 square chains 66 1 sq. furlong. 1 sq. rood. 1 sq. acre. 1 link. 1 pole. 1 chain. 1 furlong. 1 mile. 1 Gunter's chain. 1 mile. 98001 1 acre. * Some surveyors use the chains of 2 poles, others of 4 poles. 7 4. SOLID, OR CUBICAL MEASURE. By this are measured all things that have length, breadth, and thickness. 40 feet of round timber, or 50 feet of hewn timber make 42 feet make 16 cubic feet make 1 foot. 1 yard. 1 ton. 1 foot of wood, or one cord foot. 128 solid feet, or 8 feet in length, 4 feet in breadth, and 4 ft. in height, or 8 X 4 x 4 = 128= 1 cord of wood. 1 bush. strick measure. 1 bush. heaped 66 The Winchester bushel is 18.5 inches in diameter, and 8 inches deep. For convenience, the following table may be used. INTRODUCTORY COURSE ΤΟ MENSURATION. REMARKS. Ir is presumed that the pupil is already well acquainted with common arithmetic; for, without a correct knowledge of that important science, he cannot reasonably expect to make much proficiency in the higher branches of mathematics, satisfactory to himself or creditable to his instructor. But, as this may not have been attended to with sufficient care, there can be no objection to a cursory review of the most important rules used in the solution of questions in mensuration. As it is foreign to the subject to enter into an arithmetical demonstration, the several rules will first be given, as a reference, and the questions for solution follow promiscuously, as this method will have a tendency to familiarize the pupil in their use and application. DECIMAL FRACTIONS. ADDITION. RULE. Write the numbers under each other, observing to place tenths under tenths, hundredths under hundredths, &c. Be particular that the decimal points stand directly under each other, in a perpendicular line, both in the given numbers and in the sum or amount. Then perform the operation the same as in addition of integers. SUBTRACTION. RULE.-1. Write the numbers the same as integers, observing that the decimal points stand directly under each other. 2. Then subtract the same as in whole numbers, and place the decimal point in the remainder under those above. 9 MULTIPLICATION. RULE.-1. Write the multiplicand, and under it the multiplier, in the same manner as simple numbers; then multiply without regard to decimal points. 2. When the multiplication is finished, begin at the right hand figure of the product, and count off as many figures toward the left as there are decimal places in the multiplier and multiplicand, and there place the decimal point. 3. If the number of places in the product be less than the decimal places in the multiplier and multiplicand, prefix a sufficient number of ciphers, to the left of the product, to equal those of the multiplier and multiplicand, and then place the decimal point to the left of the ciphers. DIVISION. RULE.-Divide as in whole numbers, and point off as many figures for decimals in the quotient as the decimal places in the dividend exceed those in the divisor. If the quotient does not contain figures enough, supply the deficiency by prefixing ciphers. To reduce a decimal to a common or vulgar fraction. RULE.-Erase the decimal point; then write the decimal denominator under the numerator, and it will form a common fraction, which may be treated in the same manner as other common or vulgar fractions. To reduce a common or vulgar fraction to a decimal. RULE.-Annex ciphers to the numerator, and divide it by the denominator. Point off as many decimal figures in the quotient as you have annexed ciphers to the numerator. To reduce several denominations to the decimal of a higher denomination. RULE.-1. Multiply by as many as it takes of the next lower denomination to make one of the higher, adding in the denominations respectively as you multiply, until they are reduced to the lowest denomination in the question, and this is the dividend. 2. Then take one of that denomination of which you wish to make it a decimal and reduce it to the same denomination with the one above-mentioned, and this last number is the divisor. 3. Divide as in whole numbers, and the quotient is the an swer. To reduce a decimal to its proper value, or compound number to whole numbers of lower denominations. RULE.-1. Multiply the decimal by the number of parts in the next less denomination, and cut off as many places for a remainder (counting from the right) as there are decimal places in the given decimal, and there make the decimal point. 2. Multiply the remainder (that is, the decimal) by the next less denomination, and cut off à remainder as before; continue in this way through all the parts of the integer, and the several denominations standing on the left of the decimal points is the answer. REVIEW. What are fractions? What are decimal fractions? From what do they arise? Why are they called decimals? Ans. Because they decrease in a tenfold ratio, as tenths, hundredths, &c. How are decimals expressed? What is always the denominator of a decimal fraction? What is the point placed before a decimal called? Upon what does the value of a decimal depend? What is the difference between prefixing and annexing ciphers to decimals? How are decimals read? Repeat the process of addition-subtraction-multiplication-division-reduction, &c. QUESTIONS. 1. Add 12-34565, 7-891, 2-34, 14, 0011 together. 36.49795 2. Add 7509, 0074, 69, 8408, 6109 together. 3. Add 7569, 25, 654, 199 together. 4. Add 71-467, 27.94, 16:084, 98-009, 86.5 together. 5. Add 9607-84, 823-79, 07965, 74-821 together. 6. Add 19-073, 2-3597, 223, 0197581, 3478·1, 12:358 to gether. 7. Add 5-3, 11-973, 49, 9, 1-7314, 34-3 together. 8. From 125-64000 take 95.58756. 9. From 145-00 take 76-84. 10. From 14-674 take 5.91. 11. From 761-8109 take 18-9118. 12. From 171-195 take 125.9176. 13. From 480 take 245-0075. 14. 3·024 x 2.23 15. 25.238 × 12.17. 16. 007853 × .035 .000274855. 17. 007 × 0008. 21. 186513-239 304.81. |