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der, which will be the difference of the two numbers; then half the said difference added to half the sum, gives the greater of the two numbers, and the said half difference, subtracted from the half sum, gives the less number.
1. The sum of two numbers is 46, and their product is 504 ; what are those two numbers ? The sum of the numbers 46 X 46=2116 sq. of their sum. The product of ditto. 504x4=2016 four times the pro.
✓100=10 differ. of the 46:2=23 half sum.
10:2=5 half differ. +5 half diff.
23 half sum.,
--5 half differ. 28 greater number. Ans.
18 less number. Ans. 2. Bought a certain quantity of broadcloth for $573, 75cts. ; and if the number of cents which it cost per vard, was added to the number of yards bought, the sum would be 480; how many did I buy, and at what price per yard. Ans. 255yds. at 82,25cts. per yard.
3. If I lay out a lot of land in an oblong form, con taining 7 acres, 1 rood, and 10 rods, and taking just 142 rods of wall to enclose it; pray how many rods long, and how many wide is said lot?
Ans. 45 rods long, and 26 rods wide.
CASE 5.-To find the degree of light, heat, or attraction.
Note.-The effects or degrees of light, heat, and attraction, are in proportion to the squares of the distances, whence they are propagated.,
EXAMPLES. 1. Two men, A and B, are sitting in a room, the former 3, and the latter 6 feet distant from a fire; how much hotter is it at A's, than: at B's seat ? 3x3=9, & 6x6–36. Then, as 9:1::: 36°: 4, so that
A's place is 4 times as hot as B's. Ans. 2. If the earth's mean distance from the sun be 95,000 000 of miles, at what distance from him must another. body be placed, that it may receive a degree of light and heat, double to that of the earth ?
95 0000002 v
=67175144+mile. Ans. which is somewhat 2 less than the distance of Venus from the sun. 3. A ball descending by the force of gravity from the top of a tower, was observed to fall half the way in the last second of time ; how long was it in descending, and what was the height of the tower ?
The square roots of the distances are as the times, viz. As the ✓1: 2::the time of falling through: the whole required height.
Now, the V1=1, and =1,4142, from which take 1; ,4142 remains.
And, as ,4142 : 1,4142 :: 1:3,414 +sec. time of descent; the square of which is 11,6654 nearly. And the velocity acquired by heavy bodies falling near the surface of the earth, is 16 feet in the first second, 64 in the second second, 144 in the third second, &c., that is, the space fallen through (in feet] is always equal to the square of the time in 4ths of a second.
ft. sec. sq. As 12 : 16 :: 11,6554 : 186,4864=1864 feet nearly, height of the tower, Ans. .
Case 6.--Any two sides of a right-angled triangle given,
to find the other side. RULE.-Extract the square root of the sum of the squares of the two least sides, and that root is the greatest side ; for the square root of the sum of the squares of the two legs, is always the length of the hypotenuse. Extract the square root of the difference of the squares of either of the two least sides and the greatest side, and that root is the other side ; for the square root of the difference of the squares of either leg and the hypotenuse, is always the length of the other leg.
EXAMPLES. 1. A ladder 40 feet long may be so planted as to reach a window 33 feet from the ground, on one side of the street ; and, without moving it at the foot, will do the same by a window 21 feet high on the other side ; how wide is the street ?
40%=1600, 332=1089. 21=441. Then 1600-1089
-511 and ✓ 511=22,6 ; and 1600—441=1159, and ✓ 1159=34,04; then 22,6+34,04=54,64 feet. + Ans.
2. A line 27 yards long will exactly reach from the top of a fort to the opposite bank of a river, known to be 23 yards broad ; what is the height of the wall ?
Ans. 14,142+ yards. 3. Two ships sail from the same port; one sails due east 50 leagues, and the other due north 84 leagues; how far are they then apart ? Ans. 97,75+leagues.
TO EXTRACT THE CUBE ROOT:
To extract the Cube Root, is to find a number which, being multiplied into its square, shall produce the given number.
Rule.-1. Separate the given number into periods of three figures each, by putting a point over the unit figure, and every third figure both ways from the place of units.
2. Find the nearest less cube to the first period by the table of powers or trial; set its root in the quotient; suba tract the cube found from the first period, and to the remainder bring down the second period, and call this the resolvend.
3. To three times the square of the root just found, add three times the root itself, setting this one place more to the right than the former, and call this sum the divisor, Then divide the resolvend, omitting the unit figure, by the divisor, for the next figure of the root, which annex to the former, calling this last figure e, and the part of the root before found call a.
4. Add together these three products, viz. thrice the square of a multiplied by e, thrice a multiplied by the square of e, and the cube of e, setting each of them one place more to the right hand than the former, and calf the sum the subtrahend, which must not exceed the resolvend; but if it do, then make the last figure e less, and repeat the operation for finding the subtrahend.
5. From the resolvend take the subtrahend, and to the remainder join the next period of the given number for a
new resolvend; to which form a new divisor from the whole root now found, and thence another figure of the - root as before, &c.
3x32=27 · 48228,544(36,4 root.
1572544 subtrahend. 2. What is the cube root of 092727. Ans. 103. 3. What is
334965783? Ans. 327. 4. What is
Ans. ,05138.+ 5. What is
Ans.. 6. What is
Ans. ,873.+ 7. What is
Ans. 1,25. + 8. What is
79? Ans. APPLICATION AND USE OF THE CUBE ROOT.
EXAMPLES. 1. The statute bushel contains 2150,4197 + cubic or solid inches; I demand the side of a cubic box which shall contain just that quantity ?
3/2150,419724=12,907+inch. Ans. Note,- The solid contents of similar figures, are in proportion to each other, as the cubes of their similar sides or diameters.
TO EXTRACT THE ROOTS OF POWERS IN GENERAL. 179
2. If a bullet 4 inches diameter weigh 9th , what will a bullet of the same metal weigh, whose diameter is 8 inches? 4x4X4=64 8x8x8=512. As 64:9::512:7276. Ans.
3. If a solid globe of silver, of 3 inches diameter, be worth $150, what is the value of another globe of silver, whose diameter is eight inches? 3x3x3=27. 8x8x8=512. As 27: $150 :: 512 :
$2844;Ans. The side of a cube being given, to find the side of that cube which shall be double, triple, &c. in quantity to the given cube.
Rule.-Cube the given side, and multiply it by the given proportion between the given and required cube, and the cube root of the product will be the side sought.
4. If a cube of silver, whose side is 2 inches, be worth $20, what should the side of a cube of like silver be, whose value would be 8 times as much ?
2x2x2=8, and 8x8=14. 3/64=4 inches. Ans. 5. There is a cubical vessel whose side is 4 feet; I demand the side of another cubical vessel, which shall contain 4 times as much ? 4x4x4=64, & 64x4=256. 3/256=6,349+feet. Ans.
6. A cooper having a cask 40 inches long, and 32 ingat ches at the bung diameter, is ordered to make another cask of the same shape, but which shall hold just twice as much; what will be the bung diameter, and length of the new cask ? 40x40x40x2=128000; then 3 128000=50,3+ in
ches length Ans. 32x32x32x2=65536 ; & 365536=40,3+ inches
bung diam. Ans.
TO EXTRACT THE ROOTS OF POWERS IN GENERAL.
RULE.-1. Prepare the given number for extraction by pointing off from the place of units as the root required directs; 4th root put a dot over every 4th figure &c. from the place of units; 5th root, over every 5th, &c. from units' place, &c.
2. Find the first figure of the root by trial, and sub. tract its power from the given number.