ELEMENTS OF

GEOMETRY AND MENSURATION.

PART I.

GEOMETRY TREATED AS AN EXACT SCIENCE.

DEFINITIONS AND FIRST PRINCIPLES.

1. GEOMETRY* is the science which treats of the form and extent of bodies, and also of the relations which different bodies, or different parts of the same body, bear to each other in respect of magnitude, or position.

Hence it is necessarily much concerned with the terms 'length', 'breadth', 'height', 'depth', 'thickness', 'area', 'content', or 'volume'; and that which does not possess some one or more of these properties is not a subject for Geometry. For instance, to determine how many quarts of water there are in a vessel which we can measure is within the province of Geometry; but to determine what will become of the water, if a fire be placed under the vessel, is beside the subject. So likewise the size of a certain carpet is a subject for Geometry, but not the colour of the carpet, or the material of which it is made.

2. It is clear then, that much of our business, as Geometricians, is with the boundaries of bodies, (that is, with whatever bounds their extent) for upon these the magnitude of every body manifestly depends, as well as its form and other properties relating thereto.

We shall have to do with more than boundaries in certain cases; but whatever be the subject of our inquiry, it will always have reference to magnitude and position, one or both, to the exclusion of all dissimilar properties, such as colour, softness, hardness, &c.

The word Geometry strictly signifies 'land-measuring'; and is supposed to have had its origin in the necessity of re-arranging the fences or other land-marks destroyed by the periodical overflowings of the river Nile.

It is clear also that, having to treat of bodies, or parts of bodies, in respect of magnitude and position, we have to provide for taking measurements of various kinds; and hence is required a sort of geometrical language in the first onset, which must be learnt from the following Definitions:

3. We measure a distance by a 'line'; so that a line will represent any one of the dimensions length, breadth, height, girth, depth, or thickness. We do not inquire as to the thickness of the line, when used for this purpose of measurement. Hence the common

DEFINITION. A LINE is length without breadth or

thickness.

It is not meant that any line we can actually use or make is without breadth or thickness; but that for Geometrical purposes, that is, as a measure of length, the length only of a line is considered.

Thus, for illustration, if the length of a room be in question, we regard not the fact of its being measured by a broad tape or a narrow tape-even the finest thread we can use will serve our purpose, if it be inextensible, -we expect the same result in each case, because it is length only we are concerned with. In the case here supposed, the broad tape is not inferior to the finest thread; but, as there are numberless other cases in which this is not so, (as will appear hereafter), the Definition of a 'line' above given is the only one which can insure general accuracy of measurement.

4. Another term in common use in Geometry is 'point', by which is meant generally no more than a place to start from, or to stop at, in drawing or measuring a line. A point hath position only, and is nothing for us to measure; and hence the common

tude.

DEFINITION. A POINT hath no parts and no magni

It is true we cannot exhibit such a point, (because that which hath no magnitude cannot be visible to the human eye); but the more nearly the points we use in practice approach the strictness of this Definition, the more accurate, it is obvious, will be the measurements which begin or end at those points.

It follows, that each extremity of a line is a point.
5. Lines are of two kinds, straight and crooked.

A straight line, or, as it is often called, a right line, is the direct, that is, the shortest, line connecting the two extremities, or extreme points, of it.

A crooked line is not the direct line joining the two points which are its extremities. It may consist of two or more straight lines joined together thus,

or in some other way. Or it may be what is called a curved line, no part being straight, such as

or

such as may be represented by a fine thread drawn tight round the trunk of a tree to measure its girth.

In speaking of points we distinguish one from another by using the letters of the alphabet to mark their position; and so also with regard to lines to mark either their position or extent, or both.

A single letter will fix or express a point, but two are mostly used to express a straight line. Thus, if we put A at one end of a straight line, B at the other end, the points, which are the extremities of that line, would be simply called the points A and B ; and the line would be called the line AB.

Sometimes, however, a single letter may be used to denote a line, but not often.

6. SUPERFICIES, SURFACE, or AREA. These words all express the same thing, which is a subject for measurement; as, for instance, the acre-age of a field. It is obvious that this will depend upon the length and breadth of the field, but not at all upon the depth of the soil, or the thickness of the sod. And so we have the

DEFINITION. A SUPERFICIES, SURFACE, or AREA, is that which hath only length and breadth.

It is not meant that the body whose superficies, surface, or area, we are considering has only length and breadth, but that the dimensions of a superficies, surface, or area, are entirely dependent upon length and breadth, to the exclusion of thickness, height, or depth. Thus in speaking of the quantity of carpet which will cover a floor, the thickness of the carpet never enters into our consideration, but only the length and breadth.

Hence the expression superficial measure' is always

understood to exclude thickness. Thus, for instance, the area or surface of this page, that is, the space upon it capable of receiving the impression of type, is manifestly independent of the thickness of the paper.

7. SURFACES are of two kinds, plane and curved.

A plane surface is one on which a straight line may be drawn in any part of it, wholly coincident with the surface. Or, in other words, if any two points are taken in the surface, and a straight line be drawn joining the two points, that line shall be wholly in the surface.

A curved surface is one, on which if points be taken and joined by lines lying wholly on the surface, those lines are found to be curved lines.

Thus the top of a table is a 'plane surface'; but the boundary of a globe is a 'curved surface'.

Observe, it is not necessary to a curved surface that all lines drawn on it should be curved lines; there may be straight lines in particular cases. For example, the surface of a round pillar is curved, but yet the lines drawn on it in the particular direction of the length of the pillar will be straight lines, whilst all others will be curved.

8. ANGLES. A plane rectilineal angle is formed by two straight lines, which meet together, but are not in the same straight line. The angle is the measure of the inclination of the one line to the other; but how that measure is taken does not concern us at present to know. All that is here required is to know how to compare one angle with another, viz.:

(1) That the angle formed by, or between, the lines AB, and AC, which meet at the point A, is equal to the angle between the lines DE, and DF, which meet at the point D, if, when the point A is 'applied to', or placed

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upon, the point D, and the line AC upon the line DF, then also the line AB coincides with DE.

(2) That the angle between AB and AC is greater or less than the angle between DE and DF, according as, when AC is applied to DF as before, AB falls farther from, or nearer to, DF, than DE does.

9. An angle is generally denoted, or expressed, by three letters of the alphabet, in the following manner: The middle letter invariably marks the point where the lines which form the angle meet together, and of the other two letters one is upon one of the lines and the other upon the other line.

Thus, if the lines BA, BC, BD, meet together at the same point B, the angle between BA, and BD, is called the angle ABD, or DBA, whichever we please, only taking care that B is the middle letter; the angle between BA and BC is called the angle ABC, or CBA; and the angle between BD and BC is called the angle DBC or CBD.

Sometimes, however, when only two lines meet together, forming only one angle, so that no mistake can arise as to the angle meant, that angle may be described by a single letter placed at the point where the lines meet. Thus, the angle formed by two lines which meet at the point A would be called 'the angle at A’.

The point where the lines which form an angle meet together is called the angular point, or vertex of the angle; and ought to be carefully distinguished from the angle itself.

Observe, the magnitude of an angle does not at all depend upon the length of the lines by which it is formed, but only upon their position. Yet the lines must be some length to be lines at all.

10. If one of the lines which form an angle be extended in the same straight line from the angular point, so as to form a second angle on the same side of it adjacent to the former, and these angles are found to be equal (8)* to each other, then each of the angles is called

This will be the mode of referring to a previous paragraph, or article, as it is usually called. In this case, it is meant that the reader look back to the paragraph numbered 8, and see that a method has been there explained of comparing one angle with another.