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which, it is advisable that he should draw them much larger than the examples here given, so that the slightest deviation from the utmost accuracy may be at once apparent. It is not enough that he should know how such figures may be drawn, but he should be able to produce them himself at will.

31. Within a circle to draw a square, octagon, etc.—First draw a diagonal (as ab);

intersect it by another (cd) at right-angles to it, and the points, abcd, will give the corners of the square required. For an Octagon—divide one of the sides of the ascertained square (ac), as figured, and as—ec will give the measures of the sides of the required octagon. It may be here observed, that one of the readiest ways of ascertaining the accuracy of a square or rectangle (14), is to measure its diagonals. If these are found to be unequal, neither the square nor rectangle can be correct.

32. Within a circle to draw a pentagon, decagon, etc.—First divide the circumference of the circle into four equal parts, as shown in the foregoing example, then take any one of the radii, or half- ..

diameters of the circle, and divide it into two equal parts f <::\

(as at the point a) • on this point (a) place the compasses, and ../ / j extending them to c, strike the arc dcc, cutting the diam eter ab at c. Then, placing the compasses on the point c, extend them to the intersection (c), and describe the arc doe. The points, where this arc cuts the circumference of the given circle, connected to c, as figured, will give two sides of the required pentagon} which ascertained, the remaining three sides are easily defined.

A pentagon, or even one of its sides, once obtained, the process of producing upon its basis a Decagon, as shown in the second example, needs no further explanation.

33. To draw an ellipse with the compasses is extremely difficult, and the process, at best, is complicated, uncertain, and unsatisfactory; for, there are no portions of the line by which it is formed that exactly corresponds to a true arc of a circle. It has been found that there are two points on the longest diameter of an ellipse, equidistant from its extreme points, called its foci, or focuses, which, if connected by two lines meeting at the circumference, no matter to what point on the circumference they may be directed, the sum of these two lines is equal to the length, or a c longest diameter of the ellipse. Thus (as first figured), ab^bc, ad—dc, and ae—ec, will be a c c f a c

g found severally equal to the diameter f g. To ascertain these important jw/ttte: having first decided upon the length a b

and breadth of the required ellipse, as ab-cd (in the second figure), and drawn these two diameters, bisecting each other at right-angles, take the measure of one half of the longest diameter (ab) with the compasses, and from the point c, as a centre, describe an arc (abe). The points where this arc cuts the diameter a b, will be those required. Now, by placing two pins in these points, and stretching a thread between them, passing over another pin at the point c, we have, as

34. It should not be understood that the methods here given aré to be considered so far arbitrary as to exclude others in common use, that may be equally as efficient, and the student will doubtless often have occasion to exercise his ingenuity in finding ready expedients, in the course of his practice, and in none more so than in supplying the place of instruments. It is well enough, when practicable, to have all such facilities; but it is equally well to know how to do without them, especially for the £>ff-hand draughtsman, who can not always have his magazine of tools by him, and who often finds in a stout piece of paper all he absolutely needs for the spontaneous manufacture of such aids as he may require at the moment, and thus he sets to work.

35. First, for his Straight-edge, or Rider. If he does not find the edge of his paper sufficiently accurate, he folds it neatly over, runs his thumb-nail along the crease to give it sharpness and firmness, and has, at once, not only the ruler he requires, but, by folding, refolding, and a little dexterous use of his penknife, soon learns the válue of his expedients, and, in a measure, to do

it were, a moveable line, equal to the length of ab, which will accurately guide a pencil in describing the required ellipse (as figured in the third example). Or, we may regulate the length of the string at once, by the required length of the ellipse, and, by doubling it, get the position of the required points on the long diameter, as well as making it serve in the after process. Ten minutes' practice will make the operation familiar to the draughtsman.