Chap. VIII.

GEOMETRY

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CIIAP. VIII.

GEOMETKY.

Geometry, derived from the Greek words which signify land and the method of measuringit, is employed in estimating the length of a line, the area or superficial contents of a figure,and the cubical or solid contents of a body, and is usually divided into theoretical andpractical. Egypt gave birth to the study; from thence it was imported into Greece , andamong the refined and intelligent inhabitants of that classic land it arrived to a degree ofperfection that succeeding ages have but little improved.

Theoretical geometry is founded on ideas, or those perceptions of the mind whichresolve and demonstrate the truth of any proposition ; it is the method of defining exactlythe notions we form of any particular figure.

Practical geometry , so important to the civil engineer, is that division which enableshim, by the aid of various mathematical instruments, to carry into operation the principlestaught by theory, and to trace and define the boundaries of any figure that may berequired to be set out for mechanical or other purposes. This branch is subdivided intoTrigonometry , Planeometry , and Stereometry .

Trigonometry is the art of measuring heights and distances by means of triangles; forexample, it enables us to ascertain the distance between the two spires A and B, which areseparated by a wide and deep river, and con-sequently inaccessible. It shows us an easymethod of laying down the map of a country,and defining upon it the inequalities of thesurface, as the depths of valleys or the heights ofmountains; it enables the mariner to map aninaccessible coast, to form a chart which shallguide him through deep and safe channels, andto shun the rocks and shoals which would destroyhis vessel.

Planeometry is applied to ascertaining the areaor superficial contents of any surface; it shows theland surveyor the method of finding the numberof acres contained in a given district of country,or of subdividing it into any number of equalor unequal parts.

Stereometry or mensuration is the art by whichwe ascertain the contents of a cube, or any solidfigure; that is to say, the quantity of cube feetor yards it may contain.

The elements of practical geometry, or leadingdefinitions, may be thus described:—a point isthe extremity of a line without dimensions, oreven length, breadth, or thickness; hence, bysojne it is merely held as an idea ; but Euclid , the earliest and ablest teacher of the science,considered it the beginning or nucleus of all quantity. It is therefore the smallest portionof matter, or of an object, that the eye can distinguish, or which may be defined or ex-pressed by a pencil, or instrument of any kind.

A point may be established at any given place, and may be marked by a stake or staff,or by the end of a pair of compasses.

The point of junction is that where two lines meet, as atQ,* The point of intersection is where two or more linescross each other, as at A.

The point of incidence, A, is that from whence theline P A is reflected back from the line A C or B C. Apoint has neither length, breadth, nor thickness; a linehas length only; a surface has length and breadth, anda solid has, in addition, thickness: one dimension isrequired then for a line, two for an area, and threefor a solid body. The edges which bound a solidare lines, and where they unite may be consideredpoints; these may all be imaginary, but they arenecessary to establish, before we can arrive at the true

Fig. 607