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3. What is the square root of 2? Ans. 1,41421356.+ 4. What is the square root of 10342656? Ans. 3216. 5. What is

6. What is

7. What is

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9. What is

964,5192360241 ?

Ans. 31,05671.

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CASE 1.-To find a mean proportional between any two given numbers.

RULE.-Multiply the two given numbers together, and extract the square root of the product, which root will be the number sought.


What is the mean proportional between 16 and 36 ?

36x16=24. Ans.

CASE 2.-To find the side of a square equal in area to any given superficies.

RULE.-Extract the square root of the given superficies, which root will be the side of the square sought.


1. If an acre of land contains 160 square rods, what will be the side of a square, which should contain just an acre ? ✓160-12,649+ rods. Ans. 2. A general having an army of 5184 men wishes to form them into a square; how many must he place in rank and file? 5184-72. Ans.

3. Let 8192 men be formed into an oblong, so that the number in rank may be double the file.




64 in file. 64×2-128 in rank.

4. Suppose a gentleman would set out an orchard of 864 trees, so that the length shall be to the breadth as 3 to 2, and the distance of each tree, one from the other, 7 yards; how many trees must there be in length, and how many in breadth, and how many square yards of ground do they stand on?

To resolve any question of this nature, say, as the ratio in length is to the ratio in breadth, so is the number of trees to a fourth number, whose square root is the number in breadth; then as the ratio in breadth is to the ratio in length, so is the number of trees to a fourth number, whose root is the number in length. And as unity is to the distance, so is the number in length less by one to a fourth number; next do the same by the breadth, and multiply the two numbers thus found together, and the product will be the answer.

As 3:2864: 576,& /576=24 num. in breadth. Ans. As 2:3864: 1296,&/1296=36 num. in length Ans. As 1:7:: 36—1:245. And, as 1 : 7 :: 24—1 : 161. And 245×161-39445 square yards. Ans.

CASE 3.-To ascertain the proportionate capacities of water pipes.

RULE.-Square the given diameter, and multiply said square by the given proportion; the square root of the product is the answer.


Admit 10hhds. of water are discharged through a leaden pipe of 21 inches diameter, in a certain time; what must be the diameter of another pipe, that shall discharge four times as much water in the same time?

21=2,5 and 2,5×2,5=6,25 square.


4 given proportion.

✔25,00-5 inches diam. Ans.

CASE 4. The sum of any two numbers, and their product being given, to find each number.

RULE. From the square of their sum, subtract 4 times their product, and extract the square root of the remain

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 46×46=2116 sq. of their sum. The product of ditto. 504×4=2016 four times the pro. [numbers.

46-2-23 half sum. +5 half diff.

✓100=10 differ. of the
10+2-5 half differ.
23 half sum.

28 greater number. Ans.

-5 half differ.

18 less number. Ans.

2. Bought a certain quantity of broadcloth for $573, 75cts.; and if the number of cents which it cost per yard, 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 $2,25cts. per yard.

3. If I lay out a lot of land in an oblong form, containing 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?

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Ans. 45 rods long, and 26 rods wide.

CASE 5.-To find the degree of light, heat, or attraction. NOTE.-The effects on degrees of light, heat, and attraction, are in proportion to the squares of the distance, whence they are propagated.


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,& 6×6=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 95,000000 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

-=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 distance are as the times, viz. As the 1:2:: the time of falling through: the whole required height.

Now, the √1=1, and √√2=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 11,6554 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: 186,4864-186 feet nearly, height of the tower, Ans.

CASE 6-Any two sides of a right-angle 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 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.


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?

402-1600. 332=1089. 212-441. 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.


A cube is any number multiplied by its square.

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; subtract 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 call 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

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