EXAMPLES.

(6) Required the Diameter of a Circle that will comprehend within its Circumference the Quantity of an Acre of Land?

(7) In the Midst of a Meadow well stored with Grass,
I took just two Acres to tether my Horse ;

How long must the Cord be, that feeding all round,
He mayn't graze less or more than these two Acres of
Ground?

Case 4. Any two Sides of a right-angled Triangle, ABC, being given, to find the remaining Side.

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 other Side.

RULE.

From the Square of the Hypothenuse subtract the Square of the given Side, the Square Root of the Remainder gives the Side required.

EXAMPLES.

(8) At Matlock, near the Peak in Derbyshire, where are many surprising Curiosities of Nature, is a Rock by the Side of the River Derwent, rising perpendicular to

a wonderful Height, which being inaccessible, I endeavoured to measure, and found by a Mathematical Method, that the Distance between the Place of Observation and the Foot of the Rock was 55 Yards, and from the Top of the Rock to the said Place was 140 Yards (nearly). Required the Height of this stupendous Work?

(9) A Ladder 40 Feet long may be so planted that it shall reach a Window 33 Feet from the Ground on one Side the Street; and without moving it at the Foot, will do the same by a Window 21 Feet high on the other Side. The Breadth of the Street is required? (10) 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. The Height of the Wall is required?

(11) 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?

(12) Two Ships set sail from the same Port; one of them sails due East 50 Leagues, the other due North 84. How far are they asunder?

QUESTIONS for Exercise at Leisure Hours.

(13) 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 Edge 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? (14) Required the Length of a Shore, that being to strut 11 Feet from the Upright of a Building, will support a Jamb 23 Feet 10 Inches from the Ground?

(15) There are two Columns, in the Ruins of Persepolis, left standing upright; one is 64 Feet above the Plane, the other 50. Between these, in a right Line, stands an ancient Statue, the Head whereof is 97 Feet from the Summit of the higher, and 86 Feet from the Top

of the lower Column; the Base whereof measures just
76 Feet to the Centre of the Figure's Base. By these
Notices, the Distance of the Top of the Columns may
be, by Numbers, easily found.

(16) A Castle Wall, there was, whose Height was found
To be a 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?

(17) As I was walking out one Day,
Which happen'd on the first of May,
As Luck would have it, I did spy
A May-Pole raised up on high,
The which at first me much surpris'd,
Not being before-hand advertis'd
Of such a strange uncommon Sight;
I said I would not stir that Night,
Nor rest content, until I'd found
Its Height exact from off the Ground.
But when these Words I just had spoke,
A Blast of Wind the May-Pole broke,
Whose broken Piece I found to be
Exact in Length Yards Sixty-three,
Which by its Fall broke
up a Hole,

Twice Fifteen Yards from off the Pole;
But this being all that I can do,
The May-Pole now being broke in twở

Unequal Parts, to aid a Friend,

Ye Youths, pray then an Answer send.

Case 5. 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.

(18) A General disposing his Army into a Square Battle, finds he has 23716 Men; required the Number in Rank and File?

LV. The EXTRACTION of the CUBE ROOT.

To extract the Cube Root is to find out a Number, which being multiplied into itself, and then again into the Product, produceth the given Number.

As the Cube Root of 729 is 9, consequently 9x9x9=729 the given Number, and so of others, as in the following Table.

Roots. 12 31 4 5 | 6 71 8 Cube. 18 27 64 125 343512729

216

RULE.

1. Make a Point over every third Figure given, beginning at the Unit's Place; seek the greatest Cube to the first Point on the Left Hand (by the Table), whose Root place in the Quotient; then Subtract its Cube from the Period, and to the Remainder (if any) bring down the three Figures, or your next Period, and call it your Dividend.

2. Find a Divisor, by calling your Quotient Figure, with a Cipher joined to it, r; then three Times the Square of r will be your Divisor; seek how often it is contained in the Dividend, and put the Answer in the Quotient as in Division, only with this Difference: call the said Quotient Figure last put up e, and multiply your Divisor by it, and place the Produce underneath the Dividend; then multiply the Square of e by three Times r, and place it also under the Dividend : lastly, cube the Figure you call e, and place it under the Dividend then add the three Products together, gives the Subtrahend, which subtract from your last Dividend, and to the Remainder bring down the next Period, and proceed as before.

:

EXAMPLES.

(1) What is the Cube Root of 21024576 ?
(2) Extract the Cube Root of 92398647 ?
(3) What is the Cube Root of 2716243264 ?
(4) What is the Cube Root of 91 ?

(5) What is the Cube Root of 67527834239 ?
(6) Extract the Cube Root of 4764,75.

(7) The Solidity of a Cube is 36155,027576

is the Side of that Cube?

Inches, what

(8) What is the Side of that Cube which contains 67667,921875 solid Inches?

(9) What is the Cube Root of 219565329 ? (10) What is the Cube Root of 3105926,917 ? (11) What is the Cube Root of ,000421875?

(12) What is the Side of a Cube whose Solidity 28022810,390625 ?

The Biquadrate of any Number is found by extracting the Square Root of the given Number first, and then the Square Root of that Root.

(13) Let it be required to extract the Biquadrate of 4857532416.

The Root of the Square cubed, or sixth Power of any Number, is found by extracting the Square Root of the given Number, then extract the Cube Root of that Square Root, which will give the sixth Power required. (14) Let it be required to extract the Square cubed Root of 49656.

The Root of the Biquadrate squared, or eighth Power, is found by first extracting the Square Root of the given Number, which will reduce it to a Biquadrate, which proceed with as before directed.

(15) Let it be required to extract or find the Root of the eighth Power of 43046721.

The Root of the Cube cubed, or ninth Power of any Number, is found by extracting the Cube Root of the given Number, and the Result will be a cubic Resolvend: of this extract the Cube Root also, which will be the Rog of the ninth Power.