and a half of the small divisions on from the 1; which 125 being set to 10 on B, we observe that the lowest number on A is 100; and opposite thereto we see 8 on B: wherefore dividing 8 by 100 and also by 10, that is in all by 1000, the quotient is 008 the factor required.

To prove the truth of the result let any dividend whatever be named, for instance, 2150-42, and it will

To Find by the Lines A and B, the Divisor Corresponding to a Given Factor.

This being the converse of the last problem, the proportion will be, As the given factor is to unity, so is unity to the divisor required.

Let, for the sake of illustration, the divisor corresponding to 008 be demanded, and the proportion will be as follows:

on A.

on B.

on A. on B. As 008: 1 :: 1 : 125 the divisor required.

In solving this example we multiply the given factor 008 by 1000, and it becomes 8. We then set 8 on A to 1000 on B, and look opposite to 1 on A, for 125 on B.

In a similar way may be found the divisor that answers to any given factor whatever.

As the art of Gaging is a branch of the Mathematics as much depending on Geometry as on Number, we deem it proper as the next introductory step, to discuss briefly such parts of Geometry as are more fundamentally connected with the subject.

45

GEOMETRY.

GEOMETRY treats of points, lines, superficies, and solids; to have a clear conception of which, it is necessary first to define the sense applied to the terms.

GEOMETRICAL DEFINITIONS.

1.

A point is an inconceivably small portion of space, being position without any magnitude.

༠

A line is length only, and generated by the motion of a point.

3.

An area or superficies has length and breadth, but no thickness.

4.

A solid has length, breadth, and thickness.

5.

A line joining any two points is said to be straight when it is the shortest possible.

6.

A plane is a superficies in which no two points can be taken such that, the straight line joining them does not coincide entirely with that superficies.

7.

A rectilineal angle is the meeting of two straight lines in a point, with any degree of inclination to one another; as ABC, or DEF, or GHI.

A line is said to be perpendicular to another when it stands on it in such manner as to make the angle on one side equal to that on the other; as AB on CD.

A right angle is formed by two lines standing perpendicular to one another; as ABC.

A geometrical figure is any space inclosed by one or more boundaries, and is either superficial or solid-superficial when the boundary consists of a line or lines, and solid when the figure has thickness.

11.

A circle is a figure bounded by one uniformly curve line called the circumference, lying wholly in the same plane; as ABCD.

The centre of a circle is the point which, in the plane of the circumference, is equidistant from every part of the curve; as C.

C

13.

The diameter of a circle is a line drawn through the centre, and terminated both ways by the circumference; as AB.