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99.6 feet. Distance required
708.6 feet. Difference in height of land 40.58 feet. 2. Two persons made observations on the altitude of a meteor, both being on the same side of it, and in a vertical plane passing through it. The distance between their sta- . tions was 200 rods, and at one, the angle of elevation was 36° 25', at the other 32° 50', and at the last the disk of the meteor subtended an angle of 5!. Required the distance from the last place of observation, also the height and diameter of it.
M. Q. R.
Height, 3 0 70
Diameter, 45 feet 5 1-2 inches. 3. From the top of a steeple 165 feet high, the angle of depression of the nearest bank of a river is 11° 15', that of the opposite bank is 6° 151. Required the width of the river.
Answer, 41.13 rods. 4. What length of cart tire will it take to band a wheel 5 feet in diameter ?
Answer, 15 feet, 8 1-2 inches. 5. A gentleman laid out a garden in a circle, containing one acre, one quarter, and one rod, with a gravelled walk on the outer side of it within the circle, which took up twelve rods of ground. What is the diameter of the circle, and what is the width of the walk ?
Answer. The diameter 16 rods-width of the walk 4 feet.
6. Neptune laid out 1,000 square miles of the surface of the sea in' a circle, and sold to Æolus all that part of it which lies without a concentric circle of one third of the diameter. What is the diameter, and how much was sold ?
Answer. The diameter 35.68 miles,
The quantity sold 888,92 square miles. 7. A farmer laid out an elliptical orchard, the longest diameter of which was 30 rods, and the shortest was 20 rods, and enclosed the same with a wall two feet thick within the figure. What is the quantity within the wall, and how
A. Q. R.
Covered by the wall 9.3
A. Q. R.
Length of each side 45 rods.
Answer, 12 by 20 rods. 10. The area of a parallelogram is five acres one quarter and thirty five rods, and the diagonal is 43 rods. Required the length of the sides.
Answer, 35 by 25 rods. 11. Required the dimensions of a parallelogram containing twenty-six acres, one quarter, and twenty-four rods, when the length exceeds the breadth by fifty-two rods.
Answer, 44 by 96 rods. 12. Required the dimensions of a parallelogram containing 250 acres, when then the sides are in the proportion of 7 to 3.
Answer, 305 1-2 by 130.93 rods. 13. John and Jonathan divided a lot of land equally as to value, which contained 600 acres, and which was appraised at $2 per acre. John's division was valued at 50 cents per acre more than Jonathan's. What were the contents and price per acre of each division. Answer. John's division contained 263 acres and 12 rods, and the price per acre was $2.28 1-10 nearly
Jonathan's division contained 336 acres, 3 roods, and 28 rods, and the price per acre was $1.78 1-10 nearly.
14. The state of Connecticut contains a little more than 4,828 square miles, or 3,090,000 acres, including rivers, harbors, creeks, roads, &c. If this quantity is laid in a square, what will be the length of a side?
M. Q. R.
Answer, 69 1 75.11. One word of advice to the young surveyor, who is coming forward to be useful in his occupation, will close the appendix.
In the choice of assistants to perform practical operations, never call to your aid Sir Richard Rum. He frequently changes his name to brandy, gin, whiskey, &c. He is treacherous, and he causes the head to whirl, the body to real, and the foot to stumble.
By his might, the strong man has fallen, and the promising youth has been brought to an untimely grave.
If you employ Sir Richard, your columns of latitude may differ too much for correct work, and your columns of departure may be still worse. You can place no confidence in him, and it is hoped that you will too highly respect your own character to be found in such company.
Note.-Such theory as is more curious than useful, how. ever correct it may be, has been excluded from the appendix. The plainest methods have been selected. On some points, the Author has been more minute on account of the fact that many copies of this book are bought by men who do not expect to be surveyors, and who do not place themselves under instructors.
If some repetition is to be found in this work, the learner will find less fault with it than the critic.
Hebron, (Conn.) June 1835.
The learner, who for the first time becomes acquainted with the wonderful properties of Logarithms, may be not a little surprised to find himself introduced to a system of numbers, so new in their nature, and which, surpassing all his former knowledge of figures, afford so many facilities for shortening the labor and lessening the difficulty of arithmetical calculations.
He will admire to find, that by help of these, the labor of hours, and in some calculations, even the labor of days, may be reduced to as many minutes! The invention of Logarithms was justly regarded as a favor from heaven;" because, in many departments of science, essential to the happiness of man, they have saved him ages of toil.
Although it does not come appropriately into the design of a work like this, to enter minutely into the history of their invention, nor the yet more difficult process by which they were originally constructed, yet a familiar explanation of their properties and uses, adapted to the apprehension and wants of the practical surveyor, is necessary, in order to his making a proper application of their great advantages in practice.
Logarithms, then, we may first observe, never stand for the numbers themselves, of which they are composed, but invariably for other numbers, of which they are only the representative exponents, or indices. Their great utility in arithmetical operations, consists, chiefly, in this,--that addition takes the place of multiplication, and subtraction that of division. That is, to multiply numbers, we have only to add their logarithms; to divide, we have only to subtract the logarithm of the divisor from that of the dividend ; to raise a number to any power, we multiply its logarithm by the exponent of that power ; and to extract the root of any number, we merely divide its logarithm by the number expressing the root to be found.
The constant number upon which the tables in common use are constructed, and which is called the base of the tables, is 10; and every conceivable number, large or small, integral, mixed, or decimal, is considered as some ascertained power or root of 10.
Logarithms. 10, whose exponent is
1. 100, whose exponent is 2. 1,000, whose exponent is 3. 10,000, whose exponent is 4. 100,000, whose exponent is 5. 1,000,000, whose exponent is 6.
NOTE. It may be remarked, that the first power of any number, is that number once
repeated, or it is the number itself: The second power of any number, is the product of that number multiplied once by itself: The third power of a number, is the product of the number multiplied twice by itself:
The fourth power of a number, is the product multiplied three times by itself, &c. The index denoting the power, is called, in common arithmetic, the exponent of that power; and is, in other words, the logarithm of the power.
Logarithms, then, are the Exponents of a series of powers and roots.
In the above series, the logarithms indicate how many cyphers belong to their corresponding numbers. Thus, the logarithm 1 stands for 10, or 1 and one cypher; the logarithm 2 stands for 100, or 1 and two cyphers; the logarithm 3, for 1000, or 1 and three cyphers, &c. Now if we multiply 10,000 by 100, the product will be 1,000,000, whose logarithm is 6 : but to obtain this, we need only add the logarithms 2 and 4, which stand opposite the numbers to be multiplied. On the contrary, if we divide 1,000,000 by 100, the quotient will be 10,000, whose logarithm is 4: but to obtain this, we need only subtract 2, the logarithm of the divisor, from 6, the logarithm of the dividend.
Again the square of 1000, that is, the product of 1000 multiplied by itself, is 1,000,000, whose logarithm is 6; but to obtain the square of 1000, we need only double its logarithm 3. On the other hand, the cube root of 1,000,000 is 100, whose logarithm is 2; but this is obtained by dividing 6, the logarithm of the given number, by 3, the index of the root. Hence it is manifest, that the protracted labor of multiplying or dividing one large number by another, the tedious evolution of roots, and the various mistakes incident to long operations, may be almost entirely obviated by the use of logarithms.
As the logarithm 1 is always 0, and that of 10 is but 1, the logarithms of all numbers below 10, will be decimals; and as the logarithms in the common system increase regularly by 1, according to the integral powers of 10, it follows that the logarithms of all numbers between 10 and 100, will be more than 1, but less than 2—that is, they will be 1 and a decimal; the logarithms of all numbers between 100 and 1000, will be between 2 and 3– that is, they will be two and a decimal : and the logarithm of all numbers between 1000 and 10,000, will be between 3 and 4—that is, 3 and a decimal.
A logarithm generally consists of two parts ; a whole number, and a decimal. This whole number or integer is called the characteristic or index, of the logarithm, and is always one less than the number of integral fig. ures in the natural number whose logarithm is sought. As the index of the logarithm is omitted in the tables, it is important to recollect the principle, or rule, by which it is to bes upplied, whenever it is wanted in calculation. Thus, the logarithm of 8 is 0.903090. Here, the number (8) consists of but one figure, and the index of its logarithm, being one less, must be 0. Again, the logarithm of 16 is 1.204120. Here, the given number (16) consists of two figures, and the index of its logarithm, being one less, must be 1.Again, the logarithm of 640 is 2.806180. Here, the given number (640) consists of 3 figures, and the index of its logarithm, being one less, must be %, &c. The rule holds universally true, that the index of a logarithm is always one less, than the number of integral figures in the natural number whose logarithm is sought.
The same rule holds in mixed numbers. The logarithm of 6.40 is 0.806180, the same as for 640 (see the last example) differing only in the index. Here, the integral part (6) of the given number, consists of but one figure, and the index of its logarithm, being one less, must be 0. And, generally, having obtained the logarithm of any number, large or small, we have only to change the index, agreeably to the above rule, in order to obtain the logarithm of every other number, consisting of the same significant figures, whether they be integral, fractional, or mixed. Thus:The logarithm of 7596
is 3.880585 759.6