But as the Extraction of Roots higher than the Biquadrate are difficult in common Numbers, on Account of their high Involutions (or Multiplications); we will now point out an easier as well as shorter Method of extracting the Roots of all Powers and Numbers, how high and large so ever, by the Logarithms.

Rule.

Seek the Logarithm of seven, or (if possible) eight of the leading Figures of the Number given, and prefix to that Logarithm an Index answerable to the Number of Places in the whole Number, (i. e. one less than the Number of Places) that Logarithm being divided by the Index of the Power, whose Root is fought, as by 2 for the Square; 3 for the Cube; 4 for the Biquadrate; 5 for the Surfolid, &c. the Quotient will be the Logarithm of the Root required.

Example.

What is the Cubed Cube Root (or ninth Power) of 4722366482869645213696?

Operation.

We find the Logarithm of the seven leading Figures 4722366 to be 67415963, and consequently the Index to be prefixed is 21; because the given Number confifts of 22 Places. This intire Number (Logarithm and Index) 21.67415953 being divided by 9, the Index of the Cubed Cube or ninth Power, the Quotient will be 2.40823995, the Logarithm of 256, the Root required.

A COMPENDIOUS COURSE

OF

PRACTICAL GEOMETRY.

G

EOMETRY is that Part of

Mathematical Learning,

which teaches how to measure the Earth, and determine the Magnitude and Distance of all Bodies contained therein. It is of the utmost Use in Life; for here nothing can deceive us, by appearing bigger or less, higher or lower,-nearer or further off, than it really is.

This Science is of very remote Antiquity, and supposed to take its first Rise in Egypt. * The Inhabitants of that Country were in a Manner compelled to invent it, to Remedy the Confufion, which generally happened in their Lands, from the Overflowings of the River Nile, which carried away all their Boundaries, and totally effaced the Limits of their Possessions. Thus this Invention, which at first consisted only in laying out and measuring the Lands, that every Person might have what justly belonged to him, was called Geometry, and it is probable, that the Draughts and Schemes which they were annually compelled to make,

* It is generally allowed that the Chaldeans were first possessed of the Mathematical Sciences, which must imply a Knowledge of Geometry. Whether Abraham (as some learned Men think) taught these Sciences first to the Ægyptians, when he went from Ur of the Chaldees, is not clear; but on this we may depend, that the Ægyptians were the first People that cultivated Geometry, and applied it, as the Word expresses, to Land Measure: they being compelled thereto by a Kind of Neceffity, in order to afcertain every Man his legal Property and Estate, in a Country where Boundaries and Land Marks were swept away and confounded by yearly Inundations.

helped them to discover many excellent Properties of these Figures, which Speculations continued to be gradually improved, and are still improving to this Day.

From Egypt Geometry passed over into Greece, where it continued to receive new Improvements in the Hands of the Learned; * and at Length it grew into such Value and Estimation, that Plato, who flourished about 300 Years before Chrift, would admit none to his Lectures who had not made fome Advances in it, thinking them not capable or fit Hearers. Whence that famous Inscription (said to be written) over his School Doors, †-Let none ignorant of Geometry enter here.

For the Pursuit of Geometrical Speculations will not only inure us to attend closely to any Subject; to seek and gain clear Ideas; to diftinguish Truth from Falfhood; to judge justly and argue truly; but do by their own Nature more directly furnish us with all the various Rules of those useful Arts and Sciences of Life, viz. Menfuration, Architecture, Fortification, Navigation, Dialling, Perspective, Optics, Mechanics, Astronomy; and, in short, with a perfect Knowledge of (all Things measurable in) the Heavens and the Earth.

Geometrical Definitions.

DEFINITIONS are short Descriptions of Things expreffive of their several Properties.

A Point, which is the very Beginning of Magnitude, is supposed to be so small as to have no Parts; neither Length, nor Breadth, nor Thickness, as the Point A. *

(A)

Point

A Line is supposed to be made by the Motion of a Point, and hath Length, without Breadth or Thickness.

If a Line be quite ftrait, it is called a Right Line, as AB.

If the Line be bent or crooked, it is called a Curve Line, as CD.

If a Line turns backward and forward, it is called a Serpentine Line, as E F.

E Serpentine Line

If two Lines run equally distant from each other, they are called Parallel Lines, as GH and IK.

If two Lines lean or incline towards each other, they will at last ineet, which Place of Meeting is called an Angle, as A.

If one Line falls perpendicuarly on another, the Angle made by their Meeting is called a Right Angle, as at B.

Right Angle

B

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