ELEMENTS OF GEOMETRY AND MENSURATION, WITH EASY EXERCISES, Designed for Schools and Adult Classes. PART II.-GEOMETRY AS AN ART. BY THOMAS LUND, B.D. RECTOR OF MORTON, DERBYSHIRE; EDITOR OF WOOD'S ALGEBRA; FORMERLY FELLOW AND SADLERIAN LECTURER OF ST JOHN'S COLLEGE, CAMBRIDGE. LONDON: LONGMAN, BROWN, GREEN, AND LONGMANS. 1st. To try the outer edge; on any plane surface trace, by means of it, the angle BAC; extend BA in the same straight line to D. Then turn the square round the point A, so that the outer edge which before coin actly coincides with AC, the square is correct as to its outer edge; otherwise not. 2nd. To try the inner edge; proceed in the same manner, only making use of the inner edge where before the outer was used. EC By this method, also, the amount and quality of the error, if any, is ascertained. For, if the error be in defect, that is, if its angle be less than a right angle, the square will appear, in the two positions above mentioned, as in the annexed fig., where the angles D A B BAC, DAE are equal, and together fall short of two right angles by the angle CAE. Therefore the error of the square is equal to half the angle CAE. Similarly, if the error be in excess, that is, if the angle of the square be greater than a right angle, the angles drawn in the two positions of the square will over-lap each other, as in the angles BAC, DAE in the annexed fig.; so that the two angles together exceed two right angles by the D A B angle CAE; and since they are equal to each other, therefore, in this case also, the error of the square is equal to half the angle CAE. (5) The draughtsman's Triangle is simply a thin triangular piece of wood or ivory, with its sides accurately and smoothly made, so that any one of them may be used as a ruler, and two of them, as AB, BC, forming a right angle. A small hole is cut through the instrument, that it may be handled and moved along the surface of the paper or board more easily. This instrument is used, as it obviously may be, for drawing lines at right angles, or perpendicular, to other lines; and also for some other purposes, as will appear hereafter. (6) The PARALLEL-RULER consists of two flat-rulers, similar and equal in all respects, as AB, CD which are so connected together by means of two equal pieces C of brass, EF, GH, working loosely round A fixed pins in the rulers at the points E, F, G, H, that, when the rulers are separated, both their outer and inner edges are parallel to each other. It is requisite, not only that EF should be equal to GH, but also that the distance EH between the pins in one ruler be equal to FG the distance between the pins in the other. In which case the lines joining the points E, F, G, H always form a parallelogram (40); and as these points are equidistant from both the outer and inner edges in each ruler, those edges will always be parallel. Hence it is plain, this instrument may be used for drawing any number of parallel straight lines, or for drawing one or more straight lines parallel to a straight line already drawn. (7) A SCALE OF EQUAL PARTS is mostly a Flat. ruler which has its whole length divided into a certain number of equal parts, and each of these parts again subdivided into smaller equal parts, the several points of division being marked by lines across the face of the ruler. The common foot-rule is an example of a Scale of Equal Parts, its length being divided into 12 equal parts called inches, and each inch into parts of an inch. This instrument is used for comparing one length or straight line with another. 97. The above are the instruments which are in most common use for Geometrical purposes. Others more complex, will be described hereafter. It seems only necessary to observe here, that the workman or draughtsman is much to be blamed, who is content to work with faulty tools or instruments, when he is able to procure better; seeing that very small errors, will often, by multiplication, produce seriously defective results. In such instruments as the foot-rule, and square, this will be obvious to the most common understanding. N.B. The Definitions and Propositions of Part I. are all assumed in this Part. 98. PROPOSITION I. To draw a straight line on a plane surface between any two given points. (1) This is mostly done, if the given points be not too widely apart, by means of a flat-ruler, or straightedge. The ruler is placed so as to have the same edge exactly on both the points, and a fine pen, or pencil, is drawn carefully along it in contact with both the straight edge and the surface on which the line is required to be drawn. See (96). (2) But, if the given points be so far apart that the ruler is insufficient, then other modes are adopted according to circumstances. Thus, it is known that light always travels, if uninterrupted, in a straight line from one point to another, and consequently any workman iş readily able to determine a point A C intermediate to A and B, C B the two given points, which shall be in the same straight line with A and B. He places his eye at A, so as to see B, and marks a point C which appears to eclipse, or coincide with, B. Then he can join A and C, and also B and C, by means of his ruler, or straight-edge; and the thing required is done. (3) In some cases where the given points A and B are very distant, it may be necessary to lay down, by the eye, several intermediate points, C, D, E, &c., and by joining each contiguous pair, one continuous straight line will be traced from A to B. (4) Another mode, adopted mostly by sawyers, for marking out the course of the saw, is to stretch tightly between the points a thin cord which has been chalked throughout its whole length or dipped in some marking material, and then, while the ends are kept fixed, the cord is drawn a little from the surface of the wood and allowed to recoil with force back again, whereby a distinct line is traced between the two ends and sufficiently straight for practical purposes. Gardeners, bricklayers, and others also, make use of a tightly stretched cord for determining the straight line which lies between any two given points. But in all cases where a cord is used it must lie along the plane surface, on which the straight line is to be drawn, throughout its whole extent, otherwise its own weight will cause it to deviate from the straight line joining its extremities. 99. PROP. II. To draw a circle on a plane surface, about a given point in it as its centre, and with a radius equal to a given straight line. (1) This may easily be done, within certain limits, by means of the ordinary compasses. Open the legs until their extreme points exactly coincide with the extremities of the given line; then fix the foot of one leg on the point which is to be the centre, and by making the compasses to revolve round this point while the foot of the other leg is kept in contact with the surface on which the circle is to be drawn, the latter will trace out the circle required. The main thing to be attended to in this operation is, |