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Theorem 15.

Similar, or like Solids are to each other in a triplicate Proportion, or as the Cubes of their like Sides.

D

A

B

a

d
b

That is, the Solidity of the smaller Body d, is to the Solidity of the greater Body D; as the Diameter a b cubed, is to the Diameter A B cubed.

Many other Theorems might be added, but these will be fufficient for the young Geometer at present. The Reft, as they would better appear in a Treatise of Trigonometry, we have reserved for a Place in that Work.

Note. That the Side upon which any Figure stands is called its Base. That a Line drawn from its Top, perpendicular to the Base, is called its Altitude or Height. That two Right Lines cannot include any Space, or form any Superficies whatever.

The Explanation of fuch Characters as are generally used in the Solution of the following Geometrical Problems.

SIGNS & MARKS.

EXPLANATIONS.

+ plus, or more.

The Sign of Addition; as 5+2=7; that is, 5 added to 2 is equal to 7.

- minus, or less. The Sign of Subtraction; as 9-4-5; that is, 9 lessened by 4 is equal to 5.

x multiply by The Sign of Multiplication; as 6×8= 48; that is, 6 multiplied by 8 is equal

to 48.

divided by The Sign of Division; as 12÷4=3; that is, 12 divided by 4 is equal to 3.

= equal to

The Sign of Equality; as 16 ozs.=115. that is, 16 Ounces are equal to I Pound.

4

:: Proportion. The Sign of Proportion; as 3:6:: 8:16; that is, as 3 is to 6, so is 8 to 16.

32 Fraction.

52

Numbers placed like a Fraction do alfo denote Divifion; the upper Number being the Dividend, and the lower the Divisor.

12-42= 10 Shews that the Difference between 12 and 4 added to 2, is equal to 10.

10-2+3=5

Signifies, that the Sum of 2 and 3 taken from to is equal to 5.

Square Root. This Mark being prefixed to any Num

ber, fignifies that the Square Root of that Number is to be extracted.

✔3 Cube Root. Signifies, that the Cube Root of that Number is to be extracted.

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The Application of Algebra and Fluxions to the Solution of Problems in Geometry must be deferred till the Learner is acquainted with the first Principles of that Science.

PLANOMETRY; Or the Measuring of

PLAIN

SURFACES.

Surface or Superficies is that which hath Length and

A Breath, without Thick

The Measure, the Area, or the Content of any superficial Figure is said to be known, when we know how many less Squares, as suppose of Feet, Inches, Yards, Perches, &c. are contained within it.

Thus, suppose in the Figure ABCD each of its Sides is found to be 3 equal Parts, then it is evident the Number of little Squares of the fame Kind contained in it will be 9. For, dividing the Sides into the Number of Parts they contain, and drawing Lines through from Side to Side, the Number of little Squares formed within, will be the Area of the larger Figure.

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The Area or Content is always of the same Denomination with the Dimensions; that is, if the Measure be taken in Feet, the Content will be Feet; if taken in Inches, the Content will be Inches; and so of any other.

* If a Superficies be raised up, it is said to be convex; if it be hollow, it Is called concave; and if it be flat and even, it is called a Plane, or plain Superficies.

Problem 1.

To Measure a Square; that is, to find the Superficial Content or Area of a Square.

Definition. A Square hath four equal Sides and four Right Angles; as ABCD, in the Figure below.

Rule. *

Multiply the given Side by itself, and the Product is the Area required.

Example.

Suppose the Side A B of the Square ABCD to be 16.2 Inches, what is the Area ?

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Operation. 16.2 × 16.2 = 262.44, the Area in Inches.

* The Reason of this Rule is evident from a Sight of the last Figure divided into little Squares.

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