To prove the line by the compasses, observe, that the distance from 1 to 2 is equal to 2 to 4, from 5 to 10 the distance from 4 to 8, and 4 to 8, from 3 to 6.

To number on Gunter's line. Observe, that the figures 1, 2, 3, 4, 5, 6, 7, 8, 9, sometimes signify themselves simply or alone; at other times 10, 20, 30, 40, &c. Again, at other times, 100, 200, 300, or 1000, &c.

To find a number on the line, as suppose 134, For the figure 1, account one on the line; and for 3, take 3 of the largest divisions; and for 4 take 4 of the smallest divisions, and that is the point. Again, to find 750 on the line for 7 take 7 on the line, for 50 take 5 of the great divisions, and that is the point.

To find a small number on the line, as suppose 12. For 10 take 1 as before, as for 2, take two of the large divisions, and that is the point.

In measuring boards or timber, it is best to have a line of two feet long, and compasses one foot long.

Note, 1. Let the measurement be by the inch, foot, yard, pole, rod, &c. it is best to have it decimally divided, or so supposed, that is, into 10th parts.

2. That if one point of the compasses reach beyond the line in the work, remove the other point to the same figure or place on the other line.

Multiplication by Gunter's Line.

To multiply 5 by 7, set one foot of the compasses on it in the left hand line, and extend the other to 5 upwards, or towards the right hand, and with the same extent place

one foot in 7, and the other foot will fall on 35 in the right-haud line, which is the answer.

Division in Gunter's Line.

Example 1. Divide 63 by 3; extend from 3 to 1 downwards, or towards the left hand, and the extent will reach the same way from 63 to 21 the quotient.

Note. In multiplying you must always extend upwards, that 11..

II

is, from 1 to 2, 3, 4, &c. and on the contrary, in dividing, extend downwards.

2. Divide 2887, equally among 16 men: extend from 16 to 1 downwards; and that extent will reach the same way, from 2887. to 18/. for each man.

3. Suppose 750l. were to be divided among 25 men: extend from 25 to 1 downwards; and that extent will reach the same way, from 750l. to 301. each man's share.

The Rule of Three Direct.

Example 1. If 6 bushels of barley cost 11s. what will 40 bushels cost? Extend from 5 to 11 upwards: and that extent will reach the same way, from 40 to 88, the shillings required.

2. If 3 ells of holland cost 10s. 6d. what will 40 ells cost? Extend from 3 to 10 upwards; and that extent the same way will reach from 40 to 140s. the answer.

The Use in Board Measure.

If a board be 9 inches broad, and 19 feet long, what is the content in superficial square feet? Extend from 12, the centre of foot measure, to 9 downwards, and that extent the same way will reach from 89 to 14 and 1.

In Timber Measure.

A piece of timber 24 inches square, and 8 feet long, what is the content in solid feet: Extend from 12 the centre, to 24 upwards, and that extent twice the same way, will reach from 8 to 32 feet, the content.

Brick Work.

How many rods of work are there in 4085 feet? Extend from 272 downwards to 2, and that extent the same way, from 4085, will reach 15 rods, the answer.

Coggeshall's Sliding Rule.

This rule is framed 3 ways; sliding by one another as the Glazier's rule; sliding on one side of a two-feet joint rule; one part sliding on the other, in a foot of length; the back part being flat, on which are sundry lines and scales.

Úpon the aforesaid sliding side of the rule, are four lines of numbers, three are double lines, and one a single line of numbers, marked with A B C and D; the three marked A B and C, are called double lines of numbers, and figured 1, 2, 3, 4, 5, 6, 7, 8, 9. Then 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10, at the end.— That marked D is the single line of numbers, and figured 4, 5, 6, 7, 8, 9, 10, 20, 30, and at the end 40, even with and under

10, in the double line next to it; and that is called the girt-line, and so marked in the figure.

The figures on the three double lines of numbers, may be increased or decreased at pleasure; thus, 1 at the beginning may be called 10, 100, or 1000; and 2 into 20, 200, or 2000; so that when one at the beginning is 10, then one in the middle is 100, and 10 at the end is 1000; but if one at the beginning is accounted for one, then 1 in the middle is 10, and 10 at the end is 100.

And as the figures are altered, so must the strokes or divisions between them be altered in their value, according to the number of parts they are divided into; as thus, from 1 to 2 it is divided into 10 parts, and each tenth is divided into 5 parts; and from 2 to 3, it is divided into 10 parts, and each tenth into 2 parts, and so on from 3 to 5; then from 5 to 6 it is divided into 10 parts only; and so on unto 1 in the middle of the rule, or the end of the first part of the double line of numbers. The second part of the double line is divided like the first.

The girt-line, marked D, is divided from 4 to 5 into 10 parts, and each tenth into 4 parts, and so on from 5 to 10; and then from 10 to 20 it is divided into 10 parts; and each tenth into 4 parts; and so on all the way from 20 to 40 at the end, which is right against 10 at the end of the double line of numbers.

The lines on the back-slide of this rule, that slide on one side, are these, viz. a line of the inch measure from 1 to 12, each divided into halves, quarters, and half-quarters; another line of inch measure from 1 to 12, each divided into 12 equal parts, and a line of foot measure, being one foot divided into 100 equal parts, and figured 10, 20, 30, 40, 50, 60, 70, 80, 90, and 100, even with 12 on inch measure.

And the back-slide of the sliding place is divided into inches, halves, quarters, and half quarters, and figured from 12 to 24, so that it may slide out to 2 feet, to measure the length of a tree, or any thing else you have occasion to measure.

The Use of the Double Scales.

square, whose B, to 3 on the feet on the line

Example 1. Suppose there is a geometrical sides are 3 feet each: set one foot on the line line A, and then against 3 on the line B, is 12 A, which is the content of such a square. 2. Suppose the side of a rhombus be 8 feet 6 inches, and the breadth of the line A B 8 feet 4 inches, what is the content ? set one foot on the line B, to 8 feet on the line A, then again 8 feet on the line B, is 72 feet parts of a foot on the line A; and to know the value of the decimal or part of the foot, look for on the rule, and you will find against it.44 inches; so that the content of this rhombus is 72 feet 4 inches.

10

3. Suppose the length of a rhomboides to be 17 feet, or 17 , and the breadth 8 feet 7, or 8, what is the content? Set one foot on the line B, to 17,25 on the line A, then again, 8,58 on the line B, is 148 feet on the line A. The figure hath been represented before, and operated arithmetically, therefore it is here unnecessary.

4. Let the base of a triangle be 4 feet 1 inch, and the perpendicular 2 feet 1 inch: the half of the one is 2 feet 7 parts, and of the other 1 foot 7 parts. Set 1 on the line B to 4,15 on the line A theu against 1,07 half the perpendicular on the line B, is 4 feet and almost for the content. Or, if you set 1 on the line B, to 1,07 on the line A, against 4,15 on the line B, is 4 and almost foot on the line A.

5. If you set 1 on the line B, to 4,1 on the line A, then against 2,25 on the line B, is 8 feet (which is about 11 inches) on the line A, the half whereof is 4 feet 5 inches, which is the content of the triangle.

BOOK ON MENSURATION.

Hutton's Mensuration, 8vo.

PART VIII.---DRAWING.

DRAWING is the accurate representation of the colours or outlines of objects. This art forms the basis on which all the other acquirements of the artist must be built; and it is not only necessary to the student in the beginning of his career, but in proportion as the professor improves in the facility and correctness of his drawing, so much the nearer does he advance towards perfec tion in his art. For the sake of those who have no opportunity of receiving regular instructions, we shall endeavour to comprise in as small a compass as possible, such directions and rules as will be easily understood and applied to practice.

Various are the opinions upon the best modes of beginning to Tearn drawing; and it is by no means an easy matter to decide upon this point, as so much must always depend upon the genius, turn of mind, and opportunities of the student. But for general purposes, and where circumstances will admit of it, we have no hesitation in recommending to begin the study of geometry and perspective. The first forms the best introduction to a knowledge of form, by giving accurate ideas respecting the most simple bodies, of which all the others may be considered as

compounds and the last seems absolutely necessary not only to enable us to draw the representations of regular objects, but even to see them correctly; as it is certain that no one unacquainted with its rules, can ever attain the power of drawing without making the grossest mistakes.

IMPLEMENTS USED IN DRAWING.

A Drawing-board for fixing the paper upon, so that it may not shift, and also by straining it, to prevent the colours, when laid wet upon the paper, from causing it to swell up, so as to be uneven. The simplest sort is made of a deal-board, framed square, with a strong piece across each end, to prevent warping. Upon this the paper may be fixed down with pins, wafers, or sealing-wax, or strained with paste or glue.

The best kind of drawing-boards, are made with a frame and a moveable pannel, upon which the paper is put wet, and then forced into the frame, where it is confined by wedges at the back. This strains equally well, without the trouble of pasting, so that you may dry it at the fire; it is proper to observe that all the angles of drawing-boards should be exactly square.

Parallel rulers are for drawing parallel lines readily; they are made of two pieces of ebony fastened together by brass bars, so as to move parallel to each other.

T-squares are rulers made in the form of the letter T, which are used with the drawing-boards; the short end, called the stock, being applied to the edge of the board, so as to slide forward and backward, while the long part, called the blade, is used for drawing lines. These are more convenient than parallel rulers, when a drawing-board is used, as by them lines are drawn at right angles to each other at once without using the compasses. The pentagraph, this is an instrument by means of which a person may copy, enlarge, or reduce the outlines of any picture, print, or drawing,

Dividing compasses are instruments of brass and steel, for dividing lines, and laying down measures from scales, &c. They are generally sold in cases, containing also a steel pen, for drawing lines clearer than can be done by a common pen, which is useful where neatness is required, and points with a black lead pencil, for putting into the compasses, when circles are to be described. These cases also contain scales of equal parts, and protractors for laying down angles.

Black-lead pencils are of various qualities: the best are fine, without any grit, not too soft, and that cut easily without breaking. Indian rubber, or elastic gum, is a substance like leather, which has the useful property of erasing or defacing lines drawn with black-lead; it is therefore much used for this purpose.