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equally apply to the pen points of the compasses before described. The drawing-pen should be kept close to the straight edge (see STRAIGHT EDGE), and in the same direction during the whole operation of drawing the line.

For drawing close parallel lines in mechanical and architectural drawings, or to represent canals or roads, a double pen (fig. 2) is frequently used, with an adjusting screw to set the pen to any required small distance. This is usually called the road pen. The best pricking point is a fine needle held in a pair of forceps (fig. 3). It is used to mark the intersections of lines, or to set off divisions from the plotting scale and protractor (p. 33). This point may also be used to prick through a drawing upon an intended copy, or, the needle being reversed, the eye end forms a good tracing point.

A STRAIGHT EDGE.

As many instruments are required to have straight edges for the purpose of measuring distances, and of drawing straight lines, it may be considered important to test the accuracy of such edges. This may be done by placing two such edges in contact and sliding them along each other, while held up between the eye and the light: if the edges fit close in some parts, so as to exclude the light, but admit it to pass between them at other parts, the edges are not true: if, however, the edges appear, as far as the test has now proceeded, to be true, still this may arise from a curvature in one edge fitting into an opposite curvature in the other; the final step, then is to take a third edge, and try it in the same manner with each of the other two, and if in each case the contact be close throughout the whole extent of the edges, then they are all three good*.

"To draw a straight line between two points upon a plane, we lay a rule so that the straight edge thereof may just pass by the two points; then moving a fine-pointed needle, or drawing-pen, along this edge, we draw a line from one point to the other, which, for common purposes, is sufficiently exact; but, where great accuracy is required, it will be found extremely difficult to lay the rule equally with respect to both the points, so as not to be nearer to one point than the other. It is difficult also so to carry the needle, or pen, that it shall neither incline more to one side than the other of the rule; and, thirdly, it is very difficult to find a rule that shall be perfectly straight.

"If the two points be very far distant, it is almost impossible to draw the line with accuracy and exactness; a circular line may be described more easily, and more exactly, than a straight or any other line, though even then many difficulties occur, when the circle is required to be of a large radius. "And let no one consider these reflections as the effect of too scrupulous exactness, or as an unnecessary aim at precision; for, as the foundation of

Euc. bk. i. def. 10. Peacock's Algebra, 1st edition, art. 532. p. 429.

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all our knowledge in geography, navigation, and astronomy, is built on observations, and all observations are made with instruments, it follows that the truth of the observations, and the accuracy of the deductions therefrom, will principally depend on the exactness with which the instruments are made and divided, and that those sciences will advance in proportion as these are less difficult in their use, and more perfect in the performance of their respec tive operations."*

ON SCALES.

Scales of equal parts are used for measuring straight lines, and laying down distances, each part answering for one foot, one yard, one chain, &c., as may be convenient, and the plan will be larger or smaller as the scale contains a smaller or a greater number of parts in an inch.

Scales of equal parts may be divided into three kinds; simplydivided scales, diagonal scales, and vernier scales.

Simply-divided Scales.-Simply-divided scales consist of any extent of equal divisions, which are numbered 1, 2, 3, &c., beginning from the second division on the left hand. The first of these primary divisions is subdivided into ten equal parts, and from these last divisions the scale is named. Thus it is called a scale of 30, when 30 of these small parts are equal to one inch. If, then, these subdivisions be taken as units, each to represent one mile, for instance, or one chain, or one foot, &c., the primary divisions will be so many tens of miles, or of chains, or of feet, &c.; if the subdivisions are taken as tens, the primary divisions will be hundreds; and, if the primary divisions be units, the subdivisions will be tenths.

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The accompanying drawing represents six of the simply-divided scales, which are generally placed upon the plain scale. To adapt them to feet and inches, the first primary division is divided duodecimally upon an upper line. To lay down 360, or 36, or 36, &c., from any one of these scales, extend the compasses from the primary division numbered 3 to the 6th lower sub* Geometrical and Geographical Essays, by the late George by William Jones, F.Am.P.S.

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Adams, edited

division, reckoning backwards, or towards the left hand. To take off any number of feet and inches, 6 feet 7 inches for instance, extend the compasses from the primary division numbered 6, to the 7th upper subdivision, reckoning backwards, as before.

Diagonal Scales.-In the simply-divided scales one of the primary divisions is subdivided only into ten equal parts, and the parts of any distance which are less than tenths of a primary division cannot be accurately taken off from them; but, by means of a diagonal scale, the parts of any distance which are the hundredths of the primary divisions are correctly indicated, as will easily be understood from its construction, which we proceed to describe.

Draw eleven parallel equidistant lines; divide the upper of these lines into equal parts of the intended length of the primary divisions; and through each of these divisions draw perpendicular lines, cutting all the eleven parallels, and number these primary divisions, 1, 2, 3, &c., beginning from the second.

Subdivide the first of these primary divisions into ten equal parts, both upon the highest and lowest of the eleven parallel lines, and let these subdivisions be reckoned in the opposite direction to the primary divisions, as in the simply-divided scales.

Draw the diagonal lines from the tenth subdivision below to the ninth above; from the ninth below to the eighth above; and so on; till we come to a line from the first below to the zero point above. Then, since these diagonal lines are all parallel, and consequently everywhere equidistant, the distance between any two of them in succession, measured upon any of the eleven parallel lines which they intersect, is the same as this distance measured upon the highest or lowest of these lines, that is, as one of the subdivisions before mentioned: but the distance beween the perpendicular, which passes through the zero point, and the diagonal through the

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one of the subdivisions on the lowest line, is equal (Euc. vi. prop 4) to one-tenth of a subdivision on the second line, to two-tenths of a subdivision on the third, and so on; so that this, and consequently each of the other diagonal lines, as it reaches each successive parallel, separates further from the perpendicular through the zero point by one-tenth of the extent of a subdivision, or one-hundredth of the extent of a primary division. Our figure represents the two diagonal scales which are usually placed upon the plane scale of six inches in length. In one, the distances between the primary divisions are each half an inch, and in the other a quarter of an inch. The parallel next to the figures numbering these divisions must be considered the highest or first parallel in each of these scales to accord with the above description.

The primary divisions being taken for units, to set off the numbers 5.74 by the diagonal scale. Set one foot of the compasses on the point where the fifth parallel cuts the eighth diagonal line, and extend the other foot to the point where the same parallel cuts the sixth vertical line.

The primary divisions being reckoned as tens, to take off the number 46.7. Extend the compasses from the point where the eighth parallel cuts the seventh diagonal to the point where it cuts the fifth vertical

The primary divisions being hundreds, to take off the number 253. Extend the compasses from the point where the fourth parallel cuts the sixth diagonal to the point where it cuts the third vertical.

Now, since the first of the parallels, of the diagonals, and of the verticals indicate the zero points for the third, second, and first figures respectively, the second of each of them stands for, and is marked, 1, the third, 2, and so on, and we have the following

General Rule. To take off any number to three places of figures upon a diagonal scale. On the parallel indicated by the third figure, measure from the diagonal indicated by the second figure to the vertical indicated by the first.

Vernier Scales.-The nature of these scales will be understood from their construction. To construct a vernier scale, which shall enable us to take off a number to three places of figures: divide all the primary divisions into tenths, and number these subdivisions, 1, 2, 3, &c., from the left hand towards the right throughout the whole extent of the scale.

Take off now, with the compasses, eleven of these subdivisions, set the extent off backwards from the end of the first

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primary division, and it will reach beyond the beginning of this division, or zero point, a distance equal to one of the subdivisions. Now divide the extent thus set off into ten equal parts, marking the divisions on the opposite side of the divided line to the strokes marking the primary divisions and the subdivisions, and number them 1, 2, 3, &c., backwards from right to left. Then, since the extent of eleven subdivisions has been divided into ten equal parts, so that these ten parts exceed by one subdivision the extent of ten subdivisions, each one of these equal parts, or, as it may be called, one division of the vernier scale, exceeds one of the subdivisions by a tenth part of a subdivision, or a hundredth part of a primary division. In our figure the distances between the primary divisions are each one inch, and, consequently, the distances between the subdivisions are each one-tenth of an ach, and the distances between the divisions of the vernier scale each one-tenth and one-hundredth of an inch.

To take off the number 253 from this scale. In crease the first figure 2 by 1, making it 3; because the vernier scale commences at the end of the first primary division, and the primary divisions are measured from this point, and not from the zero point *. The first thus increased with the second now represents 35 of the subdivisions from the zero point, from which the third figure, 3, must be subtracted, leaving 32; since three divisions of the vernier scale will contain three of these subdivisions, together with three-tenths of a subdivision. Place, then, one point of the compasses upon the third division of the vernier scale, and extend the other point to the 32nd subdivision, or the second division beyond the 3rd primary division, and laying down the distance between the points of the compass, it will represent 253, or 25.3, or 2.53, according as the primary divisions are taken as hundreds, tens, or units

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General Rule.-To take off any number to three places of figures upon this vernier scale. Increase the first figure by one; subtract the third figure from the second, borrowing one from the first increased figure, if ne

* If the vernier scale were placed to the left of the zero point, a distance less than one primary division could not always be found upon the scale.

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