Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

tion of some fraction "into itself, as 3×3; or as being only equivalent to some immediate power, as

Find the square root of 2??
Find the square root of 3

36

Answ..

[blocks in formation]
[ocr errors]

§3. To extract the Square Root of Surd Numbers. Such numbers as have not a perfect root, or are not perfect squares, cubes, &c. are called surd numbers.

From which definition it is manifest the square root of such numbers cannot be found exactly; but by approxi mation we may come as near the truth as we please, for which this is the

Rule.

Find the root of the given number, as if it was a perfect square, and when that is done there will be a remainder, to which prefix two cyphers (as the next lower period) and so to every succeeding remainder prefix two cyphers; and proceed at every step till one more than half the proposed number of decimal places be obtained (for all the figures arising when the cyphers are prefixed are decimals) and then the rest may be found by plain Division.

The Use of the SQUARE-ROOT.

Case I.

To find a mean proportion between any two given numbers.

Rule.

Multiply the two given numbers together, and extract the square root of the product, which roots shall be a mean proportional sought.

Examples.

1. What is the mean proportional between 4 and 9 ?

Answ. 6.

2. What is the mean proportional between 16 and 36 ? Answ, 24.

Case II.

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,

Examples.

Examples.

3. If the area of a given circle is 4276.5, I demand the side of a square whose superficial content shall be equal Answ. 65.395.

thereto ?

4. Suppose I have an elliptical or irregular fish-pond, containing in surface 9 acres, 2 roods, 15 perches, and would have a square one of the same content; I desire to know how many yards each side must be?

Answ. 274.2535 yards.

5. If the content of a given circle be 160, what is the side of a square equal thereto? Answ. 12.649.

Case III.

Having the area of a circle, to find the diameter.

Rule.

As 355: 452:: so is the area to the square of the diameter.

Examples.

6. Required the diameter of a circle that will comprehend within its circumference, the quantity of an acre of land? Answ. 99.91 yards.

7. In the midst of a meadow well stored with grass, I took just two acres to tether my ass ;

How long must the cord be, that feeding all round,
He mayn't graze less or more than these two acres of
Answ. 70.6475 yards.

Case IV.

(ground?

Any two sides of a right-angled triangle, A, B, C being given, to find the remaining side.

[blocks in formation]

1. The base and perpendicular being given to find the hypothenuse.

Rule.

Square each side, add the squares together, and the square root of this sum gives the hypothenuse required. 2 If the hypothenuse and one side be given, to find the ⚫ther side.

Rule

Rule.

From the square of the hypothenuse, subtract the square of the given side, the square root of the remainder gives the side required.

8. A line 27 yards long, will exactly reach from the top of a fort, on the opposite bank of a river, known to be 23 yards broad: the height of the wall is required?

Answ. 14.1421 yards.

9. Suppose a light-house built on the top of a rock, the distance between the place of observation and that part of the rock level with the eye, and directly under the building, is given 310 fathoms; the distance from the top of the rock to the place of observation is 423 fathoms; and from the top of the building 425: the height of the edifice is required? Answ. 287.8 fathom height of the rock. 2.93156 ditto height of the light-house.

10. Two ships set sail from the same port, one of them sail'd due east 50 leagues, the other due north 84: How far are they asunder? Answ. 97.75 leagues.

11. The height of an elm, growing in the middle of a circular Island 30 feet in diameter, plumbs 53 feet, and a line, stretched from the top of the tree straight to the hither hedge of the water 112 feet; what then is the breadth of the moat, supposing the land on the other side the water to be level? Answ. 83 feet.

12. Required the length of a shoar, that being to strut 11 feet from the upright of a building, will support a jamb 23 feet 10 inches from the ground?

Answ. 26 feet, 9 inches.

13. A castle wall there was, whose height was found
To be an hundred feet from th' top to th' ground;
Against the wall a ladder stood upright,

Of the same length the castle was in height.
A waggish youth did the ladder slide,
(The bottom of it) ten feet from the side:
Now I would know how far the top did fall,
By pulling out the ladder from the wall?
Answ. 6 inches nearly.

Case V.

Any number of men being given, to form them into square battle, or to find the number of ranks and files.

Rule.

Extract the square root of the number of men given, will give the number of men either in rank or file.

Examples.

1

Example.

14. A general disposing his army into a square battle, finds he has 23716 men: required the number in rank and file? Answ, 154 men.

THE EFFECTS OF LIGHT AND HEAT.

The effects or degrees of light, heat and attraction, are reciprocally proportional to the square of their distance from the centre whence they are propagated.

1.. Suppose that in a room, where two men, A and B are sitting, there is a fire, from which A is 3 feet, and B is 6 feet distant: it is required to find how much hotter it is at A's seat than at B's? Answ. A's is 4 times as hot as B's.

2. Supposing the earth to be 81 millions of miles distant from the sun: I would know at what distance from him another body must be placed so as to receive light and heat double to that of the earth? Answ. 57275649 miles.

3. The distance between the earth and sun is accounted 81 millions of miles, the distance between Jupiter and the sun 424 millions of miles, the degree of light and heat received by Jupiter, compared with that of the earth is re quired? Answ. The sun's influence on the earth, to that on the planet Jupiter, is as 27 to 1.

[ocr errors]

4. Mercury the nearest of all the planets to the sun, is about 32 millions of miles from him; Saturn is distant about 777 millions of miles; what proportion is there between the solar influences on these two bodies?

Answ. The solar influence on Mercury to that of Sa turn, is as 589 to 1 nearly.

5. Suppose with Dr. Keil, the distance of the sun to be from us 115 of his diameters: how much hotter is it then at the surface of the sun, than under our equator?

Answ, 13225 degrees hotter.

The less porous a Body is, the greater is its Density.

6. The compactness or density of the moon is to that of the earth, as 1324 is to 100: what proportion then is there between the quantity of matter in the earth and that in the moon, since the earth's diameter is 7970 miles, and that of the moon 2170? Answ. The earth contains 40,117

times more matter than the moon.

VELOCITIES acquired by heavy BODIES falling. The velocity acquired by heavy bodies falling near the surface of the earth, is 16 feet in the first second, and as

Bb

161

[ocr errors]
[ocr errors]

16 feet are to the square of one second, or 1, so is the given distance to the square of the seconds required, or, on the contrary, to determine what space a heavy body has passed in any time given is,

By multiplying 16 the descent of a heavy body in one second of time, by as many of the odd numbers beginning from unity, as there are seconds in the given time, viz. by 1 for the first, 3 for the second, 5 for the third, 7 for the fourth, &c. the sum total will give the-space it hath passed.

7. Suppose a stone let fall into an abyss should be stopped at the end of the eleventh second after its delivery, - what space would it have gone through?

Answ. 1946.083 feet.

8. 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: required the tower's height, and the whole time of descent?

Answ. 187
{

feet, tower's height.

3 seconds, time of descent.

9. What is the difference between the depth of two wells, into each of which should a stone be dropped at the same instant, one will meet with the bottom at six seconds, the other at ten? Answ. 1029.3 feet.

10. If a stone be 19 seconds in descending from the top of a precipice to the bottom, what is the height of -the same? Answ. 1019 fathoms, 1 foot, 84 inches. 11. In what time would a musquet ball, dropped from top of a steeple-400 feet high, be at the bottom?

Answ. 5 seconds nearly.

12. If a hole could be bored through the centre of the earth, in what time would a heavy body let fall from its surface, arrive at its centre? Answ. 18 minutes, 55 seconds, 33 thirds.

VIBRATIONS OF PENDULUMS.

It hath been found by experiment, that a pendulum 39.2 inches long, in our latitude, vibrates 60 times in i minute; and that the length of the pendulums are to one another reciprocally, as the square of the number of their vibrations made in the same space of time.

1. What difference is there between the length of a pendulun. that vibrates half a second, or 120 times in a minute, and another that swings double seconds, or 30 times in a minute? Ans. 147 inches.

2. What

« ΠροηγούμενηΣυνέχεια »