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 the diagonals AC, AD, AE as before; in AB take Ab equal to the given base; through b draw bc parallel to BC meeting AC in c; draw cd parallel to CD meeting AD ind; 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 b to make the legs CA, CB invariable, and also Ca, Cb; at the same time permitting motion round itself like the hinge of ordinary compasses.-C being the centre of this screw or hinge, A Ca is a straight line, and so also is BCb. When the instrument is used, the two limbs are first brought into exact juxtaposition, so that the points A and B coincide, and also a and b. Then, according to the requirements of the problem in hand, the centre C is fixed, making CA and CB bear a certain proportion to Ca and Cb. (This is done by means of a graduated scale on the instrument itself). Having thus fixed the legs in a certain proportion, any distance AB will bear the same proportion to ab, since ACB, a Cb are always similar triangles. So that, opening the one pair of legs to embrace any given 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. A B 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 đặt sa but we can divide the line into any even number of egna pars bsuccessively taking the half of the last bat and the griposed number of parts is attained. There is such a single form of the instrument, called -Wnoves and Fulars 18 Prop. LIII To explain the construction and use of the PANTAGRAPE The PANTAERAP s an instrument used by draughtsmen for coming drawings, (that is, for constructing emmar figures cut the same scale, or a reduced scale, or an enlarged scase, as may be required; and when we say that in good hands, it performs its work in each case with all table accuracy, without the aid of either ruler or commosses, the value of the instrument will at once be admitted. It has also this great merit, that it does not confine used 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. 4. E, and so that the lines joining the centres of these hinges (the dotted lines) form a paraliclogram ADFE, Thus the Timbs of the instrument have free motion round points 4, D, F, E, but ha eircumstances FE 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 p 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,ˆ 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, P &c. is similar to PQR &c. (71); and it is on the proposed distance in the compasses, at a guess, somewhat greater than the 5th part of AB, step along this line, beginning from C, marking the points of division so that CD= DE-EF=FG=GH. Join HB, and produce it to meet CI in the point I. Then join ID, IE, IF, IG, intersecting AB in the points d, e, f, g; and AB shall be divided by these points as required. : For, since ICD, IAd are similar triangles, by (71) Ad CD: Id: ID. Similarly de: DE: Id: ID, .'. Ad: CD: de: DE; and, alternately, Ad: de CD: DE (74 Cor. 3). But CD-DE, .. Ad=de. In the same manner it may be shewn, that de = ef = fg = gB. (3) Another still more ingenious method, which avoids entirely the drawing of any parallel line, is as follows: From one extremity A of the given line draw an indefinite straight line, making any acute angle with AB; along which make six equal steps with the compasses, marking the 4th, 5th, and 6th, by the let- ping along AB from either A or B, the given line is divided as required. For, joining BD, since ED = DC, and EB=BF, .. the sides EC, EF of the triangle ECF are divided proportionally in the points D and B; and .. BD is parallel to CF. Again, in the triangle ABD, since GC is parallel to BD, AG: AB AC AD (70 Cor.); but |