VT is the tangent of the arch FV, and its supplement

CT is the secant of the arch F V.

AI is the co-tangent, and C I the co-secant of the arch FV.

The chord of 60°, the sine of 90°, the versed sine of 90°, the tangent of 45, and the secant of 0.0, are all equal to the radius.

It is obvious, that in making use of these lines, we must always use the same radius, otherwise there would be no settled proportions between them.

of

Whosoever considers the whole extent and depth geometry, will find that the main design of all its speculations is mensuration. To this the Elements of Euclid are almost entirely devoted, and this has been the end of the most laboured geometrical disquisitions of either the ancients or moderns.

Now the whole mensuration of figures may be reduced to the measure of triangles, which are always the half of a rectangle of the same base and altitude, and, consequently, their area is obtained by taking the half of the product of the base multiplied by the altitude.

By dividing a polygon into triangles, and taking the value of these, that of the polygon is obtained; by considering the circle as a polygon, with an infinite number of sides, we obtain the measure thereof to a sufficient degree of accu

racy.

The theory of triangles is, as it were, the hinge upon which all geometrical knowledge turns.

All triangles are more or less similar, according as their angles are nearer to, or more remote from, equality.

The similitude is perfect, when all the angles of the one are equal respectively to those of the other; the sides are then also proportional.

The angles and the sides determine the relative and absolute size, not only of triangles, but of all things.

Strictly speaking, angles only determine the relative size; equiangular triangles may be of very unequal magnitudes, yet perfectly similar.

But, when they are also equilateral, the one having its sides equal to the homologous sides of the other, they are not only similar and equiangled, but are equal in every respect.

The angles, therefore, determine the relative species of the triangle; the sides, its absolute size, and, consequently, that of every other figure, as all are resolvable into triangles.

Yet the essence of a triangle seems to consist much more in the angles than the sides; for the angles are the true, precise, and determined boundaries thereof: their equation is always fixed and limited to two right angles.

The sides have no fixed equation, but may be extended from the infinitely little to the infinitely great, without the triangle changing its nature and kind.

It is in the theory of isoperimetrical figures* that we feel how efficacious angles are, and how ineffica cious lines, to determine not only the kind, but the size of the triangles, and all kinds of figures.

For, the lines still subsisting the same, we see how a square decreases, in proportion as it is changed into a more oblique rhomboid: and thus acquires more acute angles. The same observation holds good in all kinds of figures, whether plane or

solid.

Of all isoperimetrical figures, the plane triangle and solid triangle, or pyramid, are the least capacious; and, amongst these, those have the least capacity, whose angles are most acute,

Isoperimetrical figures are such as have equal circumferences.

But curved surfaces, and curved bodies, and, among curves, the circle and sphere, are those whose capacity are the largest, being formed, if we may so speak, of the most obtuse angles.

The theory of geometry may, therefore, be reduced to the doctrine of angles, for it treats only of the boundaries of things, and by angles the ultimate bounds of all things are formed. It is the angles which give them their figure.*

Angles are measured by the circle; to these we may add parallels, which, according to the signification of the term, are the source of all geometrical similitude and comparison.

The taking and measuring of angles is the chief operation in practical geometry, and of great use and extent in surveying, navigation, geography, astronomy, &c. and the instruments generally used for this purpose are, quadrants, sextants, theodolites, circumferentors, &c. as described in the following pages. It is necessary for the learner first to be acquainted with the names and uses of the drawing instruments; which are as follow.

The geometry of planes and position relate only to length and breadth. The geometry of solids to length, breadth, and thickness. This latter branch is best studied by the young reader by inspecting the various figures in wood made occasionally for that purpose. EDIT.

purposes, some to facilitate operation, and others to promote accuracy.

It is the business of this work to describe these instruments, and explain their various uses. In performing this task, a difficulty arose relative to the arrangement of the subject, whether each instrument, with its application, should be described separately, or whether the description should be introduced 'under those problems, for whose performance they were peculiarly designed. After some consideration, I determined to adopt neither rigidly, but to use either the one or the other, as they appeared to answer best the purposes of science.

As almost every artist, whose operations are connected with mathematical designing, furnishes himself with a case of drawing instruments suited to his peculiar purposes, they are fitted up in various modes, some containing more, others, fewer instruments. The smallest collection put into a case, consists of a plane scale, a pair of compasses with a moveable leg, and two spare points, which may be applied occasionally to the compasses; one of these points is to hold ink; the other, a porte crayon, for holding a piece of black-lead pencil.

What is called a full pocket case, contains the following instruments.

A pair of large compasses with a moveable point, an ink point, a pencil point, and one for dotting; either of these points may be inserted in the compass→ es, instead of the moveable leg.

A pair of plain compasses somewhat smaller than those with the moveable leg,

A drawing pen with a protracting pin in the upper part.

A pair of bow compasses.

A sector.

A plain scale.

A protractor.

A parallel rule.

A pencil.

The plain scale, the protractor, and parallel rule, are sometimes so constructed, as to form but one instrument; but it is a construction not to be recommended, as it injures the plain scale, and lessens the accuracy of the protractor. In a case with the best instruments, the protractor and plain scale are always combined. The instruments in most general use are those of six inches; instruments are seldom made longer, but often smaller. Those of six inches are, however, to be preferred in general, before any other size; they will effect all that can be performed with the shorter ones, while, at the same time, they are better adapted to large work.

Large collections are called, magazine cases of instruments; these generally contain:

A pair of six inch compasses with a moveable leg, an ink point, a dotting point, the crayon point, so contrived as to hold a whole pencil, two additional pieces to lengthen occasionally one leg of the compasses, and thereby enable them to measure greater extents, and describe circles of a larger radius. A pair of hair compasses.

A

pair of bow compasses.

A pair of small parallel ink pens.

A pair of triangular compasses.

A sector.

A parallel rule.

A protractor.

A pair of proportional compasses, either with or without an adjusting screw.

A pair of wholes and halves.

Two drawing pens, and a pointril.

A pair of small hair compasses, with a head similar to those of the bow compasses.

A knife, a file, key, and screw-driver for the compasses, in one piece.

A small set of fine water colours.