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the slider along the slit until the centre of the pin coincideswith the other extremity, and the instrument will be set inthe ratio required.

As this problem is essential to most of the following, thereader is requested to understand it thoroughly before he pro-ceeds farther.

Problem II. Figure 2.— Given any number of lines, a b,CD, e f, G II, SfC., to find a series of other lines, which willhave the same ratio in every two corresponding lines, givenone line a b, of the series required, corresponding to A b of theseries given.

Set the proportional compasses in the ratio of A b to a b byProblem I.; then if a b be greater than a b, take the distance,C d, with the remote extremities, and make c d equal to thedistance contained between the near extremities, and a b,will have the same ratio to c d, which Ab has to CD : in likemanner, take the distance, E F, with the remote extremities,and make ef equal to the distance contained between thepoints of the near extremities, and c d will have the sameratio to e f that c d has to e f ; or a b will have the sameratio to ef, that a b has to e f, and thus the remaining lines,gh, &c., will be found in the same manner ; but if the line,a b, in the given series, be less than a b, in the given line ofthe series required, the lengths, CD, if, &c., must be takenwith the shorter ends of the compasses, and the lines, c d, efi&c., made respectively equal to the corresponding distancescontained between the extremities of the longer ends.

Problem III.— To divide a given line in the same propor-tion as another given line is divided.

Figure 3.—Let ax be a given line, divided into the parts,

A b, b c, C d, dx, and let it be required to divide the line, a e,in the same proportion.

Set the compasses in the ratio of the whole lines A x anda e, then take the distances, ab,bc,cd,ds, with the samelegs which were applied in taking the distance ax, and withthe other legs corresponding to the distance a e, make ab,b c,cd,de, respectively equal to the distance between the pointsof these legs, and ab,bc,cd,de, will have the same ratioto each other, that a b, b c, C d, d x, have to each other.

Problem IY.— To divide a straight line, a x, No. 2, in con-tinued proportion ; given the extreme part, A b.

Set the compasses in the ratio of a x to b x by Problem I.;contract the points of the longer legs to bx, then, with theshorter legs, cut off the distance x C ; again contract the dis-tance between the points of the longer legs to x c, thus found,and with the shorter legs cut off the distance x D; and thus,by setting the distance between the points of the longer legsto the last distance found, the shorter legs will give the suc-ceeding part by transferring the distance between the shorterlegs from x towards A. By continuing this operation, asmany points may be found as are necessary for the pur-pose required ; then ab:bc: cd:de:ef:fg, &c.; thatis, A b : bc::bc: cd; and bc: Cd::cd: de, & c.

Any question in the rule of three may be resolved by theproportional compasses and a plane scale. Thus, supposethree articles of the same kind to cost two shillings ; whatwill eight cost ? Set the compasses in the ratio of 2 to 3 hyProblem I. ; extend the points of the shorter legs to 8 onthe scale : then apply the longer legs to the scale, and 12,the measure indicated, will be the fourth proportional, asrequired.

Where there is a great disproportion in the terms of theratio, it would be more eligible to use two scales ; thus, sup-pose it were required to find a fourth proportional to the num-bers 30, 3, and 45, take any convenient scale for the antece-dents, 30 and 45, and any convenient larger scale for theconsequents, 3 ; then, by Problem I. set the compasses in the J

~■ — — 1 ■ . . ’ ' Afl IV

ratio of the distance 30, on the scale of the antece 45the distance 3 in the scale of the consequents ; then pliedfrrnn tliA snnlft nf tliA jintPAAffontfl witli thA 1 a<7S .. 4/l

! dist* 1

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to the same scale, and the other legs will give the4.5, or 4-|> the answer as required, by applying thebetween their points on the scale of the consequents-By this, any proportion, whether in lines or nutnhebe resolved. B grid

Problem V. Figure 4.— Given two straight lin es > .^pgstC D, tending to an inaccessible point, and a point, E> ^ S 0tion, to draw a straight line through the last-named V isthat all the three straight lines, a b, C d, and thatrequired to pass through the given point, may havepoint of concourse. B 9 t

Through the given point, e, draw f g, meeting a „dand c d at G ; draw h i parallel to f g, cutting A 15 a th®c d at 1 : set the proportional compasses in the ratl ® th® 11two straight lines, f g and h i ; then, f g being g’'® a j £h 1 , take the distance G e, with the longer legs, and j e gs jequal to the distance between the points of the shor .y t iiedraw a straight line through the points e and K 5 th j,aV estraight lines a b, cd, and e k, be produced, they woone common point of concourse. j, e t' v °

Corollary. Figure 5.—Hence, if A b and fn astralstraight lines, and if e, g, i, l, be several points in a iiYline, p o, meeting a b in p, and c d in o ; then, “ ra ' at g t ad“line, r s, parallel to p O, cutting a b at r, and ^ ® 0 to £ s!setting the proportional compasses in the ratio ot 0 j, O 1 '(po being longer than rs,) take the distancesOE, 0 - ne d b®'with the longer legs, and apply the distances c0 ? g to *’tween the points of the shorter legs respectively tr . fl g tb 0from s to h, from s to k, and from s to M; and eoSO tnoQlines, e f, g h, ik, l m, they would' all have on 1

point of concourse, if produced with a b and C D- aD d lt . !

Figure 12 shows the application of this pro j) 0 rizoO tacorollary, in drawing the representations of t ‘® in p •

lines which regulate the heights of doors and win va0 j s liiOeperspective representation of a building, when 1 . tur e, a® -points of the sides, a k and A h, are not in the are jjitwhen all the points, through which these lines P a ^ ^ ■

corner of the building, or any other convenient ' A b a,l jProblem VI. Figure 6.— Given two stra T anv numb 0 r% JCD, tending to an inaccessible point, and ” ^ d rat0 ,y \points, e, G, 1 , in position, not in a straight line ’fling to 1 \

right line through each of the points E, G > ^ !

same point with A b and CD. _ throng^ p j

Through the point E, draw any line, A ’ at &n<‘ ^ |point G, draw l m, parallel to A c, cutting a \\ e \ to £ (gat m; and through the point 1 , draw N O, also P‘^ p» ra 6 , q;cutting a b at n, and c d at o; draw any l ,I,e ' 0 f aC t0A C; set the proportional compasses in the rat . S) and sntake the distances c m, c o, with the longer ? ne( j bet ^

Q r, Q s, respectively equal to the distances con a jj e l to ^the points of the shorter ends ; draw R T, s ’ c tjvely ^"^icl 1cutting a b in t, U; make Q e, r g, s i, respe^ ^ ^ G g,

C E, mg, O 1 ; and draw the straight lines, E ,will tend to the same point with A b and C • proFigure 7 is added to show the great use 0 , g p r o Zde b1m perspective, and completely exemphne pject 10 , , in gsshowing, that if any section of the origiu ^ ie m° u ‘ y l0 e3the plane of the picture be obtained, sue 1 ‘ t0 dr*W ( j, eon the face of a building, it will be very tend .y

from all the points of such a section ^ sad 1 ®

proper vanishing point, two lines tendm 0 mH 011 ’

tai KS'vn.

a" b" c" d' e f", of the end of a right cy