2. Suppose your age to be 15y. 19d. 11h. 37m. 45s. how many seconds are there in it, allowing 365 days and 6 hours to the year? Ans. 475047465. Ans. 1 year. 3. In 31536000 seconds how many years? 4. How many minutes from the first day of January to the 14th day of August, inclusively? Ans. 325440. 5. How many days since the commencement of the Christian Era? 6. How many minutes since the commencement of the American war, which happened on the 19th day of April, 1775? 7. How many seconds between the commencement of the war, April 19th, 1775, and the independence of the United States of America, which took place the 4th day of July, 1776* ? 9. LAND OR SQUARE MEASUre. 1. In 29 acres, 3 roods, 19 poles, how many roods and perches? Acres. R. Poles. 29 3 19 Proof. 4/0)477/9 4 4)119-19p. 119 roods. 40 Answer 4779 perches. 2. In 1997 poles how many acres? 29ac. 3 roods. Ans. 12a. 1r. 37p. 3. In 89763 square yards how many acres, &c.? Ans. 18a. 2r. 7p. 101ft. 36in. 4. How many square feet, square yards, and square poles, in a square mile? Ans. 27878400 feet, 3097600 yards, and 102400 poles. 10. SOLID MEASURE. 1. In 15 tons of hewn timber how many solid inches? 15 tons. 50 750 feet. 1728 6000 1500 5250 750 Proof. 50 1728)1296000(750 15 tons. 8640 Ans. 1296000 inches. 2. In 9 tons of round timber how many inches? Ans. 622080. 3. In 25 cords of wood how many inches? Ans. 5529600. Grindstones are usually sold by the solid foot, and the contents are found by the following Rule ; Multiply the sum of the whole diameter and of the half of the diameter, by the half diameter, and this product by the thickness, and you have the contents in cubic inches. 4. What is the content of a grindstone, whose diameter is 32 inches and its thickness 3 inches? 5. How many solid feet in a grindstone, whose diameter is 40 inches and thickness 4 inches? Ans. 23 feet. Note. This rule is not designed to give the solid contents with perfect accuracy. For the true rule, see Mensuration, Art. 30, 11. WINE MEASURE. 1. In 9hhds. 15galls. 3qts. of wine how many quarts? bhds. gal. qts. 9 15 3 63 32 55 Proof. 9hhds.-15gals. 582 gallons. 4 Ans. 2331 quarts. 2. In 12 pipes of wine how many pints? Ans. 12096. Ans. 9p. 1hhd. 22gal. 3qts. 4. In 1008 quarts of cyder how many tons? 12. ALE OR BEER MEASURE. 1. In 29hhds. beer how many pints? hhds. Proof. Ans. 12528 pints. 2. In 47bar. 18gal. of ale how many pints? 13. DRY MEASURE. 1. In 42 chaldrons of coals how many pecks? Chaldrons. 42 32 84 126 1344 bushels. 4 Ans. 13680 Proof. 128 64 64 Ans. 5376 pecks. 2. In 75 bushels of corn how many pints? 3. In 9376 quarts how many bushels? Ana. 4800. FRACTIONS. Parts of a thing are expressed by figures, as well as whole things. When a whole is expressed by figures, the number is called an integer. But when a part, or some parts of a thing, are denoted by figures, as one fourth, two thirds, four sevenths, three tenths, &c. of a thing, the expressions of these parts by figures are called Fractions. The term, fraction, is derived from a Latin word, which signifies to break, as an integer or unity is supposed to be broken or divided into a certain number of equal parts, one or more of which parts are denoted by the fraction. Thus one fourth denotes one of the four equal parts, and three tenths denotes three of the ten equal parts, into which a thing is broken or an integer divided. Fractions arise naturally from the operations of Division, when the divisor is not contained a certain number of times exactly in the dividend. For the remainder after the division is performed, is a part of the dividend which has not been divided; the divisor being the number of parts into which the integer is divided, and the remainder showing the number of those parts expressed by the fraction. Thus 4 is contained in 9, two and one fourth times, and, hence the quotient cannot be fully expressed in such cases, except by a whole number and a fraction. Fractions are divided into two kinds, Vulgar, and Decimal. VULGAR FRACTIONS. Vulgar Fractions are expressions for any assignable parts of unit, or whole number; and are represented by two numbers placed one above another, with a line drawn between them, thus: 4. &c. signifying five eighths, four thirds. The figure above the line is called the numerator, and that below it the denominator. The denominator shews how many parts the integer is divided into; and the numerator shews how many of those parts are meant by the fraction. Fractions are either proper, improper, single, compound, or mixed. 1. A single or simple fraction is a fraction expressed in a simple form; as, 76, &c. 2. A compound fraction is a fraction expressed in a compound form, being a fraction of a fraction; as of, of of, which 5 are read thus, one half of three fourths, two sevenths of five elevenths of nineteen twentieths, &c. 3. A proper fraction is a fraction whose numerator is less than its denominator; as ., &c. 4. An improper fraction is a fraction, whose numerator exceeds its denominator; as,, &c. 5. A mixed number is composed of a whole number and a fraction, as 73, 35, &c. that is, seven and three fifths, &c. 6. A fraction is said to be in its least, or lowest terms, when it is expressed by the least numbers possible. 7. The common measure of two, or more numbers, is that number which will divide each of them without a remainder: Thus, 5 is the common measure of 10, 20 and 30; and the greatest number, which will do this, is called the greatest common measure. 8. A number, which can be measured by two, or more numbers, is called their common multiple: And, if it be the least number, which can be so measured, it is called the least common multiple ; thus, 40, 60, 80, 100, are multiples of 4 and 5: but their least common multiple is 20. Note. The product of two or more numbers is a common multiple of those numbers. Thus, 3x4x5=60, and 60, or 3x4x5, is evidently divisible, without remainder, by each of those numbers. And the same must be true in every similar case. 9. A prime number is one, which can be measured only by itself or a unit, as, 3, 7, 23, &c. 10. A perfect number is equal to the sum of all its aliquot parts.* An aliquot part of a number is contained a certain number of times exactly in the number. PROBLEM I. To find the greatest common measure of two, or more, numbers. RULE. 1. If there be two numbers only, divide the greater by the less, and this divisor by the remainder, and so on, always dividing the The following perfect numbers are all which are, at present, known. 6 28 496 8589869056 2305843008139952128 8128 33550336 2417851639228158837784576 9903520314282971830448816128 This and the following problem will be found very useful in the doctrine of fractions, and several other parts of Arithmetick. The truth of the rule may be shewn from the first example: For, since 108 measures 216, it also measures 216+108, or 324. Again, since 108 measures 216 and 324, it also measures 5×324+216, or 1836. In the same manner it will be found to measure 2X1836+324, or 3996, and so on. It is also the greatest common measure; for suppose there be a greater, then, since the greater measures 1836 and 3996, it also measures the remainder 324 ; and since it measures 324 and 1836, it also measures the remainder 216; in the same manner it will be found to measure the remainder 108; that is, the greater measures the less, which is absurd; therefore, 108 is the greatest com ion measure. In the same manner, the demonstration may be applied to one or more al. ditional numbers. |