slides be set to 5 on the same scale, the distances between the points will be in the ratio of 1 to 5, and the area of the copy pricked off by the points D and E will be 4th of the area of the drawing, of which the lines are taken off by A and B conversely, if the lines of the drawing be taken off by the points D and E, the points A and B will prick off a copy, of which the area will be 4 times or 5 times as great, according as the line across the slider is set to the division marked 4 or 5 on the scale. To take the Square Root of a Number.—The line across the slider being set to the number upon the scale of plans, open the points A and B to take the number from any scale of equal parts (see page 9), then the points D and E applied to the same scale of equal parts will take the square root of the number. Thus, to take the square root of 3, set the line across the slider to 3, open out the compasses, till a and B take off 3 from any scale of equal parts, and the points D and E will take off 1.73, which is the square root of 3 from the same scale of equal parts. A mean proportional between two numbers, being the square root of their product, may be found by multiplying the numbers together, and then taking the square root of the product in the manner explained above. The numbers of the scale of solids are the cubes of the ratios of the lengths of the opposite ends of the compasses, when the line across the slider is set to those numbers; so that, when this line is set to the division marked 2 upon the scale of solids, the distance between the points A and B will give the side of a solid of double the content of that, of which a like side is given by the distance of the points D and E when the line is set to 3, the respective distances of the points will give the like sides of solids, the contents of which will be in the proportion of 3 to 1;. and so on. The Cube Root of a given number may be found by setting the line across the slider to the number upon the scale of solids, and, opening the points A, B, to take off the number upon any scale of equal parts, the points D, E, will then take off the required cube root from the same scale. THE TRIANGULAR COMPASSES. One of the best forms of these instruments is represented in the annexed figure. abc is a solid tripod, having at the extremity of the three arms three limbs, d, e, and f, moving reely upon centers by which they may be placed in any po a sition with respect to the tripod and each other. These limbs carry points at right angles to the plane of the instrument, which may be brought to coincide, in the first instance, with any three points on the original, and then transferred to the copy. After this first step two of these points must be set upon two points of the drawing already copied, and the third made to coincide with a new point of the drawing, that is, one not yet copied : then, by placing the two first points on the corresponding points in the copy, the third point of the compasses will transfer the new point to the copy. Another form of triangular compasses is represented in the annexed figure. B THE DRAWING-PEN. This instrument is used for drawing straight lines. It consists of two blades with steel the 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. ail 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. 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 1 40 دن GO 234 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 4 5 11 2 4 56 7 8 10 1 2 3 4 6 | |2|3| * Geometrical and Geographical Essays, by the late George Adams, editad by William Jones, F.Am.P.S. 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 ame point, being nothing on the highest line, and equal to |