being taken for another, on any return to it. But in general, the name of a mark is very easily had by referring in the book to the line it was made in. After the small alphabet is gone through, the capitals may be next, the print letters afterwards, and so on, which answer the purpose of so many dif ferent letters; or the marks may be numbered.

The letter in the left hand corner at the beginning of every line, is the mark or place measured from; and, that at the right hand corner at the end, is the mark measured to; but when it is not convenient to go exactly from a mark, the place measured from, is described such a distance from one mark towards another; and where a mark is not measured to, the exact place is ascertained by saying, turn to the right or left hand, such a distance to such a mark, it being always understood that those distances are taken in the chain line.

The characters used, are [ for turn to the right hand, for turn to the left hand, and a placed over an offset, to shew that it is not taken at right angles with the chain line, but in the line with some straight fence; being chiefly used when crossing their directions, and it is a better way of obtaining their true places than by offsets at right angles.

When a line is measured whose position is determined either by former work (as in the case of producing a given line, or measuring from one known place or mark to another) or by itself (as in the third side of a triangle) it is called a fast line, and a double line across the book is drawn at the conclusion of it; but if its position is not determined (as in the second side of a triangle) it is called a loose line, and a single line is drawn across the book. When a line becomes determined in position, and is afterwards continued, a double line half through the book is drawn.

When a loose line is measured, it becomes absolutely necessary to measure some line that will de

termine its position. Thus, the first line a h, being the base of a triangle, is always determined; but the position of the second side hj, does not become determined, till the third side jb is measured; then the triangle may be constructed, and the position of both is determined.

At the beginning of a line, to fix a loose line to the mark or place measured from, the sign of turning to the right or left hand must be added (as at j in the third line ;) otherwise a stranger, when laying down the work, may as easily construct the triangle hjb on the wrong side of the line ah, as on the right one but this error cannot be fallen into, if the sign above named be carefully observed.

In choosing a line to fix a loose one, care must be taken that it does not make a very acute or obtuse angle; as in the triangle p Br, by the angle at B being very obtuse, a small deviation from truth, even the breadth of a point at p or r, would make the error at B, when constructed, very considerable; but by constructing the triangle pBq, such a deviation is of no consequence.

Where the words, leave off, are written in the field-book, it is to signify that the taking of offsets is from thence discontinued; and of course something is wanting between that and the next offset,”

MR. KEITH'S IMPROVED PARALLEL SCALE.

Mr. Thomas Keith, teacher of the mathematics, has considerably improved the German parallel ruler, or that of Mr. Marquois, see pages 26 and 27. By making the hypothenuse, and the perpendicular line to it from the opposite angle, in the ratio of 4 to 1, and adding several scales, &c. its uses are consider ably extended for drawing plans of fortifications, and other branches of the mathematics.

Fig. 2, 3, 4, and 5, plate 33, represent the two faces of the ruler, and the triangle of half the real dimensions.

The slider is a right-angled triangle, the perpendicular is divided into inches and tenths, and the base has three indices, and other divisions requisite to be used with the scale.

The ruler contains 16 different scales, which, by the help of the slider may be increased to 20, without guessing at halves and quarters. The figures on the one end shew the number of divisions to an inch, and the letters on the other end are necessary to exemplify its use. It likewise contains the names of the polygons, the angles at their centres, and a scale of chords by which all polygons may be readily con

structed.

ITS USE AS A COMMON PARALLEL RULER.

po

To draw a line parallel to the slider. Lay the hypothenuse, or sloped edge of the triangle in the sition you intend to have your line, place the scale against the base of the triangle, and draw a line along the slope edge, keep the scale fixed, and move the slider to the left or right hand, according as you want a parallel line above or below the other.

To draw a line parallel to the scale. Lay the scale in the position you intend to have your lines, and draw a line along the edge of it. Place the base of the triangle against this edge, the middle index standing at O on the scale, and make a mark against any one division on the perpendicular; turn the triangle the other side uppermost, and make a mark against the same division; join these marks, and this line will be parallel to the former. The same may be done by sliding the triangle, without turning it, if the lines are not required to be very long.

To draw a line parallel to the slider at a great distance. Draw a line along the slope edge of the triangle, and fix the scale as before, make a mark against 2 on the perpendicular; the index being at O, turn the triangle the other side uppermost, and

make a mark against the same division, take away the triangle, and move the scale to these marks, apply the triangle again; proceed thus till you have got the proper distance, then draw a line along the slope edge, and it will be parallel to the former.

To draw a line parallel to the scale at a great distance. Draw a line along the edge of the scale, place the base of the triangle against that edge, the middle index being at O, and make a mark against 2 on the perpendicular; turn the triangle the other side uppermost, and make a mark against the same division; take away the triangle, and move the scale to these marks; proceed thus till you have got the proper distance, then draw a line along the scale, and it will be parallel as required.

ITS USE IN ERECTING PERPENDICULARS.

If a line is wanted perpendicular to the scale, apply the base of the slider to it, and draw a line from the scale along the perpendicular of the triangle; should a longer perpendicular be wanted, apply the perpendicular of the triangle to the scale, and draw a line along the base.

If a perpendicular is wanted to a line, which has been drawn along the hypothenuse of the triangle, keep the scale fixed, and apply the hypothenuse of the triangle to it, then draw a line along the perpendicular of the triangle.

ITS USE IN CONSTRUCTING POLYGONS.

Having made choice of any one scale, take the side of your polygon from it. Take the degrees under the name of your polygon, and subtract them from 180; at each end of the above side of your polygon, make angles equal to half the remainder, and the distance from the intersection of the lines, which form the angles, to either end of the side of your

polygon, will give the radius of its circumscribing circle.

Or, the three angles, and one side being given of a triangle, the radius, or remaining sides, may be found by the rules of trigonometry.

ITS USE IN DRAWING PARALLEL LINES AT ANY GIVEN DISTANCE, WITHOUT THE ASSISTANCE OF A PAIR

OF COMPASSES.

If the side of your polygon was taken from the scale C.. Move the slider from 0 to 1, 2, 3, &c. on the scale D, and you will draw a parallel line of the width of 1, 2, 3, &c. divisions on the scale C. This scale is 20 fathoms to an inch.

In a similar manner the scales G, L, and P, are to be used.

If the side of your polygon was taken from the scale C, calling the divisions two each. Move the slider from O, on the scale A, to 1, 2, 3, &c. on the same scale, and you will draw a parallel line of half the width of 1, 2, 3, &c, divisions on the scale C, This scale is 40 fathoms to an inch.

Similar instructions may be applied to the scales G, L, and P.

If the side of your polygon was taken from the scale C, calling the divisions three each. When the middle index stands at O, the divisions marked C on the slider will make straight lines with 5 and 6 on the scale D. By moving the slider to the right or left till the other divisions thereon make straight lines successively with 5, 6, &c. you will draw a parallel line of one-third of the width of 1, 2, 3, &c. divisions on the scale C. This scale is 60 fathoms to an inch.

Similar directions must be observed in using the scales G, L, and P.

If the side of your polygon was taken from the scale A. Move the slider from O, to 1, 2, 5, &c. on