RULE. To the earth's diameter, (41815224 feet,) add the height of the eye, and multiply the sum by the height of the eye. The square root of the product is the distance at which an object ON THE SURFACE of the earth or water can be seen. Work in the same way with the height of the object, and the sum of the two results is the distance at which the object may be seen. How far may a mountain, that is 11 miles high, be seen from the mast-head of a ship, 50 feet above the surface of the water ? v (41815224 + 50) x 50 45724 ft. or 8 ms. v (41815224 + 7920) x 7920 = 575534 ft. or 109 Ans. 117 ms. ms. 4. How far can Bunker Hill Monument, which is 282 feet above the level of the sea, be seen from the deck of a ves. sel, the spectator's eye being 15 feet above the water ? 5. How far may a mountain 2} miles high, be seen from the mast-head of a vessel, 40 feet above the water ? PROBLEM III. To determine the distance of a gun, or a thunder cloud, from seeing the flash, and hearing the report. RULE. Multiply the number of seconds that elapse between the flash and the report by 1142, for the distance in feet. 6. Four and a half seconds after seeing the flash of a cannon, the report was heard. What was the distance ? 7. What is the distance of an electrical cloud, if the thunder is heard in 23 seconds after the flash is seen? PROBLEM IV. To find the pressure of water against the banks of a stream or the dam of a pond. RULE. Multiply the area of the bank by one half the depth of the water, for the cubical contents of a column of water equivalent to the pressure. 8. The gate of a floom is 18 feet deep and 16 feet wide. What pressure does it sustain ? 9. What amount of pressure is sustained by a bank whose area is 5694 feet, the average depth of water being 10,5 feet? QUESTIONS FOR REVIEW. EVERY PRINCIPLE INVOLVED IN THE FOLLOWING QUESTIONS SHOULD BE FULLY AND CLEARLY EXPLAINED BY THE PUPIL. What is ARITHMETIC? What is a number? In how many ways are written numbers expressed ? How many figures are employed for the purpose ? What are they called? Why are they so called ? What is a unit? What is an abstract number !-an applicate number? What is an integer ?—a fraction? Of how many operations does Arithmetic consist? What are they called ? What is the object of each ? What is NUMERATION? How many modes of numeration are now in use ? Describe the Roman method ? What is the leading principle of the Arabic method ? How many characters are employed to represent numbers ? What does zero represent? What is its use? What is the decimal point? How many places are embraced in a period ? Repeat the numeration table for integers—for decimals. * How are decimals read? What is the effect of zeroes at the right of decimals? How may this be shown? Write and explain on the board, a number containing seven periods of integers, and twenty places of decimals. How many figures were embraced in a period, in the ancient English system of Numeration ? * The table may be continued to any extent. The denominations above duodecillions, to the twenty-second period, are : Tredecillions, Quatuordecillions, Quindecillions, Sexdecillions, Septendecillions, Octodecillions, Novemdecillions, Vigintillions. What is ADDITION? How is the sign plus written, and what does it denote? How is the sign of equality written? How are numbers written in Addition? Propose an example in Addition, and perform it on the board, giving the reason for each step of the process. How may Addition be proved? What is SUBTRACTION ? What is the Minuend ?-the Subtrahend ?-the Remainder? What other terms are applied to the Remainder ? What is the form of the sign minus, and its use? How are numbers written in Subtrac. tion? If any figure of the Subtrahend exceeds the one above it, what may be done? Show that the true result may thus be obtained. How is Subtraction proved? Propose an example in Subtraction, and perform it on the board, giving the reason for each step. What is MULTIPLICATION! What is the Multiplier ?the Multiplicand ?—the Product ? What are the factors of a number? What is a prime number ?—a composite number? What is the sign of Multiplication? How is multiplication performed? How proved ? How many decimals are pointed off in the product ? Explain the reason for this. How may any number be multiplied by 10, 100, 1000, &c. ? What other abbreviations may be adopted ? How may the product be obtained in a single line? Propose and explain an example in Multiplication. What is DIVISION? What is the Divisor?—the Dividend? —the Quotient ?-the Remainder ? What is a Fraction ? What is the Numerator ?—the Denominator? How may the remainder of any division be expressed in the form of a fraction? How is division performed? How many decimals are contained in the quotient? How is division proved ? What abbreviations may be adopted ? How is the true quotient figure obtained ? What is a multiple ?-a common multiple ?–a sub-multiple? How is the least common multiple found? How may it be found by a table of prime fac. tors ? What is a common divisor, or common measure? How is the greatest common divisor found? How may it be found by a table of prime factors? Propose and explain an example in Division ;—in finding the least common multiple ;—the greatest common measure. In how many different ways may Fractions be regarded ? Show the application of each. What are the terms of a fraction? What is a proper fraction ?-an improper fraction ?-a mixed number?-a compound fraction ?-a complex fraction? How may we reduce an improper fraction to a whole or mixed number?-a whole number to a fraction having any given denominator ?-a mixed number to an improper fraction ?-a compound fraction to a simple one?a fraction to a decimal ?-a decimal to a fraction ?.—a fraction to its lowest terms ?-two or more fractions to a common denominator? Propose and explain an example of each reduction. Propose and explain examples in Addition, Subtraction, Multiplication, and Division of Fractions. What are CIRCULATING DECIMALS, and whence do they arise ? What is the repetend ?—the finite part? How may we reduce infinite decimals to fractions ? How may we divide by any number of 9's? How, when all but the units' figure are 9's? Give an example of each division, and ex. plain the principle on which it depends. What are COMPOUND NUMBERS? How are the operations on them performed ? Propose and explain an example in reducing higher denominations to lower ;-lower denomina. tions to higher ;-lower denominations to the fraction of a higher;—to the decimal of a higher. Propose and explain an example in Compound Addition ;-Compound Subtraction ;- Compound Multiplication ;-- Compound Division. What are Duodecimals? Perform examples in Multiplication and Division of Duodecimals, and explain each principle involved. Prove an example in each of the simple rules, by CASTING Show the application of the principle to errors of transposition. How may business operations in which Compound Num. bers are involved, be frequently abbreviated ? Give an example? Propose examples of abbreviations in multiplying by any number of 9's ;-in multiplying and dividing by 5;by 25;—by 75;—by 125;—by 375;-by 625;—by 875;in multiplying by any number within 12 of 100, 1000, &c.; -in squaring a number ending in 5;—in multiplying two numbers in which the tens are alike and the sum of the units is 10;-in finding the product of two numbers, one of OUT THE NINES. which is as much above, as the other is below, a certain number of tens ;-in multiplication, when one figure of the multiplier is an aliquot part of some of the remaining figures. What is PERCENTAGE? Give and explain an example of the application of percentage to Commission ;-Insurance;Taxes ;-Stocks -Gain and Loss ;-Duties;-Simple Interest ;—Compound Interest ;-Discount. What is the distinction between Simple and Compound Interest? What is the Principal ?—the Rate ?—the Amount ? What is the usual rate? What is the Bank Rule for computing interest? —the Decimal Rule? Explain the principles on which each is founded. Give an example of an Account Current. How do we find the rate, when the principal, interest, and time are given ?—the time, when the principal, interest, and rate are given ?—the principal,—the time, rate and interest being given ?—the principal,—the time, rate and amount being given ? What is the usual mode of computing discount ? Propose and explain examples in EQUATION OF PAYMENTS, and AVERAGE. What is Ratio? What is a proportion? What are the terms of a proportion called ? What are the antecedents ?The consequents ?— The extremes ?—The means ? What is said of the product in a proportion ? Prove the fact. When one of the extremes, and the two means are given, how may the other extreme be found ? How may the antecedents and consequents be diminished ? State the Rule of THREE, and solve a question by it. Explain the distinction into multipliers and divisors ;-into cause and effect. What simple rule is founded on these distinctions ? How is the statement made when the terms are fractional ? Why? What is Arbitration of Exchange? What is the Chain Rule? Show the connection of cause and effect, in the Chain Rule. Propose and explain examples in Proportion, and Arbitration of Exchange. What is the object of the rule of FELLOWSHIP? Propose and explain examples to which the rule is applicable. What is ALLIGATION? How may any desired mixture be made, when there is no limit ?—when the whole quantity is limited ?—when one or more of the ingredients is limited ? |