or value of 17., which being multiplied by the annuity 231., will give 242.42, or 2427. 8s. 4 d. the value.

Rule. For finding the value of the decimal proportion of a pound in shillings, pence, and farthings (see page 18). See Inwood's Tables on Valuing Estates, &c.

Explanation of the algebraic signs or symbols generally used to shorten productive numbers of figures, in all sorts of calculations.

Algebra is a general computation, in which abstract quantities are represented by letters and their connexion pointed out by means of certain characters or symbols. It is one of the most important branches of mathematical science.

=

+

32

33

signifies equality, 9 added to 4 is equal to 13.

plus or more, as 9 added to 4 is equal to 13.
minus or less, as 4 from 8 is equal to 4.
plus or multiply by, as 9 by 5 is equal to 45.
division, as 45 divided by 9 is equal to 5.
the
square root, as 9 is equal to 3.
the cube root, or 3rd power, as 64 equal 4.
that 3 is to be squared, as 3 is equal to 9.
that 3 is to be cubed, as 33 is equal 27.
the bar, or vinculum, denotes that two or
more are to be taken together, as
4×6+3=36, and

√62-32 = 5 denotes that 3 squared, subtracted from 6 squared, and the square root extracted, is equal to 5.

that 45 is to be divided by 9; this character is commonly used in vulgar fractions. Also the two signs may be used in like manner, thus:

--

6 2 4
= .5.

=

4 X 2 8

4 6 8 12 called proportionals; as 4 is to 6, so is 8 to 12.

25

PRINCIPLES OF GEOMETRY.

GEOMETRY, originally signified according to the etymology of the name, the art of measuring the earth, but it is now the science that treats of and considers the properties of magnitude in general. It is divided into two parts, theoretical and practical.

Theoretical geometry considers and treats of first principles abstractedly. Practical geometry applies these considerations to the various purposes of life. By practical geometry many operations are performed of the utmost importance to society and the arts.

Now the whole numeration 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, or the whole base by half 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 an approximation.

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 these angles are nearer to, or more remote from, equality.

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

The angles and sides determine the relative and absolute size of a triangle.

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

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

The angles, therefore, determine the relative species of the triangle, the side 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 sum is always fixed, and equal 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 from the theory of isoperimetrical figures* that we feel how efficacious angles are, and how inefficient lines, to determine not only the kind but the size of the triangle 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 acquire more acute angles.

The same observation holds good in all kind 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.

But curved surfaces and curved bodies, and among curves the circle and sphere are those whose capacities 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, as it treats only of the boundary of figures, and by angles the ultimate boundary of all figures are formed; the angles give them their figure.

Angles are measured by the circle; parallel lines are the source of all geometrical similitude and comparison.

* Isoperimetrical figures are such as have equal circumferences.

Taking and measuring 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 hereafter.

DEFINITIONS.

The first definition in geometry is a point, which has position, but no parts or dimension.

Note.-In surveying, a point or station is of importance, as between the beginning and end of a line there are frequently several points or stations, at which other lines commence or end; on the accuracy in fixing these points in position or in a straight line the whole survey depends.

A surface or superfices is that magnitude or quantity which is comprehended under two dimensions, length and breadth, without any regard to depth, and, therefore, bounded by lines only; so when the measure of a field, building, &c., is made, it is taken for the surface only.

A cubic or solid body is that which is contained under three dimensions; length, breadth, and thickness, or depth, is when we measure a block of stone, wood, quarry excavation, embankment, &c.

Lines are either parallel, oblique, perpendicular, or tangential; parallel lines are those which have no inclination to each other and never meet, though ever so far produced.

Note. In surveying and plotting, to draw or pole out a straight line is of the utmost importance; however simple it may appear, it is difficult. If the base lines of a survey are not most accurately poled out straight, no matter how accurate they may be chained, it will affect the whole survey, as all other lines that terminate or intersect it will be either too short or too long. The same applies also to plotting the survey on paper; the straight edge or rule should always be tested by drawing a fine pencil line on the paper, and reversing the edge.

Angles are either right, acute, or obtuse.

A figure of three sides is called a triangle, and receives particular denominations from the relation of its sides and angles.

An equilateral triangle is that whose sides and angles are all equal.

An isosceles triangle is that which has two of its sides and angles equal, and one side greater or less.

A scalene triangle is that whose three sides and angles are all unequal.

A right angle triangle is that which has one right angle, or one line, perpendicular to another.

In a right angle triangle the side opposite to the right angle is called the hypothemese, the other sides the base and perpendicular.

An obtuse angled triangle has one obtuse and two acute angles.

An acute angled triangle has all its three angles acute.

A figure of four sides and angles is called a quadrangle or quadrilateral.

A parallelogram is a quadrilateral having both its opposite sides parallel, and all the angles right angles.

A rectangle is a parallelogram whose opposite sides are equal and parallel to one another, and all the angles right angles.

A square or tetragon is an equilateral rectangle having all its sides equal and parallel to one another, and all its angles right angles.

A rhombus is a parallelogram whose sides are parallel and equal in length to each other, and the opposite angles equal to each other, forming two obtuse angles and two acute angles.

A rhomboid is a parallelogram whose opposite sides are parallel and equal to one another, also the opposite angles are equal to each other.

A trapezoid has two sides parallel to each other, and of dif ferent lengths, both perpendicular to one side; it has two right angles, the other two angles are obtuse and acute.

A trapezium is a quadrilateral whose sides and angles are all unequal to one another.

A diagonal is a line joining any two opposite angles of a quadrilateral.

Plane figures having more than four sides are called polygons,