ADE; and upon ae the triangle aef similar to AEF. Then the whole figure abcdef shall be similar to ABCDEF; and it is constructed upon the base ab. For, since each of the triangles in abcdef is similar, and equiangular to the corresponding triangle in ABCDEF, it is easily seen that the whole figures ABCDEF, and abcdef are themselves equiangular. Also, the sides about equal angles are proportional, because these sides are the sides of similar triangles, and ... as such are proportional (71). (2) Or, if the straight line, which is to be the base, be given in magnitude only, and not in position, draw E F b B the diagonals AC, AD, AE as before; in AB take Ab equal to the given base; through b draw be parallel to BC meeting AC in c; draw cd parallel to CD meeting AD in d; de parallel to DE meeting AE in e; and ef parallel to EF meeting AF in f. Then Abcdef is the figure required. The proof is obvious from (71 Cor. 2). 177. PROP. LXXVIII. To explain the construction and use of the PROPORTIONAL COMPASSES. To facilitate the construction of similar figures a very useful instrument has been invented, called ' Proportional Compasses'. It consists of two parts exactly equal, which are worked to a fine point at both ends, and are so fastened together, by means of a screw, (as seen in the annexed diagram) that they become a sort of double compasses, Aa being one of these parts, and Bb the other. Both limbs of the instrument have an equal groove or slit, in which the screw C moves, and in any position of C within this groove it can be firmly tightened, so as to make the legs CA, CB invariable, cording to the requirements of the pro- b B length or line, we have the required length on the reduced scale at once determined by the other pair; and the same may, of course, be done for any number of lines which are required to be in the same proportion. There are several other uses to which this valuable instrument is put. For instance, it has a graduated scale upon it called 'circles'; and this enables us so to fix the point C, that the circumference of the circle, whose radius is AB, shall be divided into any proposed number from 6 to 20 inclusive, of equal parts by stepping round it with the opening ab. We can do this, because there exists an invariable proportion between the side of a regular polygon of a given number of sides and the radius of the circumscribed circle (91). And, generally, whenever an invariable proportion is known to exist between two particular geometrical magnitudes of a class, it is obvious that the Proportional Compasses may in such case be usefully employed. COR. If C be irremoveably fixed, so that CA and CB are each double of Ca and Cb; then in all cases ab will be half of AB, and we can bisect any given straight line, not too long, with great ease and accuracy. Not only so, but we can divide the line into any even number of equal parts, by successively taking the half of the last half, until the proposed number of parts is attained. There is such a simple form of the instrument, called 'Wholes and Halves'. 178. PROP. LXXIX. To explain the construction and use of the Pantagraph. The PANTAGRAPH is an instrument used by draughtsmen for copying drawings, (that is,. for constructing similar figures) upon the same scale, or a reduced scale, or an enlarged scale, as may be required; and when we say that, in good hands, it performs its work in each case with all attainable accuracy, without the aid of either ruler or compasses, the value of the instrument will at once be admitted. It has also this great merit, that it does not confine itself to straight lines and circles. Any curved line whatever presents no obstacle to its working So that the most crooked fence, once laid down on paper by the surveyor, can be copied exactly as it is, or on a smaller or larger scale, with ease and accuracy, at a single operation. The best way to get a satisfactory knowledge of the instrument (and the same may be said of most other instruments) is to see it, and to see it at work. But it may be tolerably well understood from the following description of it, and of the principle of its construction. AB, AC are two bars, or rulers, and DF, EF are two shorter ones, connected together, as shewn in the annexed diagram, by means of hinges at A, D, E, F, and so that the lines joining the centres of these hinges (the dotted lines) form a parallelogram ADFE. Thus the limbs of the instrument have free motion round the points A, D, F, E, but under no circumstances can ADFE cease to be a parallelogram. When the instrument is used, it is laid flat upon the drawing board (like the ordinary parallel ruler), and it moves upon small castors placed beneath the points A, B, C, F. DB and DF are divided into the same number of parts, and the points of division are marked by figures to enable the draughtsman to set the sliding index, which is upon each of them, so as to produce the copy on the exact scale required. The sliding index is fixed by means of a clamped screw in each case, as at O and p; OpP is a straight line. Then, if the drawing is to be made on a reduced scale, a tracer is placed in a socket at P in the ruler AC, and a pencil in like manner at p; O is made the fulcrum round which the whole instrument moves, and is the only fixed point in it. The original drawing is then placed under the tracer P, and as this tracer is steadily made to traverse the outline of the drawing, the pencil p, which is in contact with the drawing-board, accurately traces out the required copy. If the copy is to be on a greater scale than the original drawing, it is only necessary for the tracer and pencil to exchange places. And if the copy is to be on the same scale, the pencil and the fulcrum exchange places. That Р must trace out a figure similar to that gone over by P will appear thus: ADFE is always a parallelogram; .. Dp is always parallel to AP; and .. Op: OP :: OD: OA. But 0, D, and A are fixed points, ... OD: OA is a fixed ratio; and Op OP is a fixed ratio, never varying throughout the operation. R Suppose then the tracer, P, to move over a straight line PQ, during which time the pencil p traces out pq; then since Oq: OQ:: Op: OP, pq is parallel to PQ. Again let the tracer move over QR, while the pencil traces out qr; since Or OR :: Op : OP Oq OQ, qr is parallel to QR. And so on to the end of the drawing; .. pqr, Q 19 P &c. is similar to PQR &c. (71); and it is on the proposed scale, since each part of the perimeter pqr &c. : the corresponding part of PQR &c. :: Op: OP. 179. PROP. LXXX. To construct a triangle whose area shall be to that of a given triangle in a given ratio. Let ABC be the given triangle; and a : b the given ratio. Find AD a fourth propor tional to b, a, and AB, so that AD is to AB in the given ratio (77). Join CD; and the triangle ACD shall be the triangle required. For, since the areas of triangles between the same parallels, that is, of the same altitude, are proportional A D E to their bases (73), the triangle ACD : triangle ABC :: AD : AB :: a: b. Or, if CE be drawn perpendicular to AB, and EF be taken a fourth proportional to b, a, and CE, so that EF: EC in the given ratio, and AF, BF be joined, the triangle ABF is the required triangle. For EAF: CAE:: EF: CE, and EBF: CBE :: EF: CE (73), .. ABF : ABC :: EF : EC :: a : b (80). H d 180. PROP. LXXXI. To construct a square whose area shall be to that of a given square in a given ratio. Let abcd be the given square, and e f the given ratio. Upon any straight line take AB equal to e, and BC equal to f. Upon AC, as a diameter, describe a semicircle e; from B draw I BD at right angles to AB, meeting the circumference in D. Join AD, CD. In DC take DE=ab, and through E draw EF parallel to AC. Then upon A E G B DF construct the square DFIH, and DFIH is the square required. For, if G be the point where EF meets BD, since EDF is a right angle, DFG, DEG are similar tri |