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j as follows.” “ On a describe the arch cm l, &c. e equal tothe circumference of the base of the cone, which divide intoeight equal parts, at the points m, l, k, i, &c. and draw thelines am, a l, a It, &c. Draw b 1, parallel to d c, and dividei c into four equal parts. Make a 5, a 11, each equal toa 4 ; make a 6, a 10, each equal to a 3 ; make a 7, a 9, eachequal to a 2; make a 8 equal to a 1. Through the points11, 10, 9, 8, and 7, 6, 5, trace the curves e 8 and 8 c; thenthe figure c 8 e i c is the superficies of the side.”

What has the division of the line i c to do with the prin-ciple? This equality is not founded upon any part of theconstruction of the solid, and consequently the method cannever obtain a true cover or envelope; indeed, it is so voidof science as not to deserve any farther notice. In the nextplace, he attempts to find the envelope of a part of a semi-cuneoid, contained between two concentric cylindric surfaces,or of the covering or lining of the soffit of a window turnedupon a centre, which has either an elliptic or circular section,

I everywhere parallel to its end, and to coincide with thej superficies of a circular wall, or the head of an aperture! j splaying on the sides, and level at the crown. But are we, j now to expect that this will be accomplished, unless by acci-! dent, when he has already failed in obtaining methods for| the description of much more simple envelopes, viz. for cylin-i ders and cones cut obliquely ? The reader will, however,

■ attend to his description, which is as follows, and then

judge for himself.

j The. descriptions of the following diagrams are equallydeficient in method, and void of principle : his diagrams,

! also, are full of redundant lines. He begins the text withoutannouncing the purpose of the operation, so that the readeri must be kept in the dark till the conclusion.'j Plate X. Figure 1. “Of straight, circular, and elliptical

i arches in circular walls. —The first work to be done, is the

; making of the centres, to turn these kinds of arches upon,which maybe thus performed: Let GHimbe the plan ofa circular building, and at Figure 6, it is required to makea centre for a semi-circular arch to the window, whose diameterwithout is ad, and within n m. Bisect a d inf, and describethe semi-circle apd. Divide ad into any number of equalparts at the points 6, 4, 2, &c. and draw the ordinates 6, 6 ;4, 4 ; 2, 2, &c. Divide n m into the same number of equali, parts, and make the ordinates 6, 5 ; 4, 3; 2, 1, &c. equal to

| the ordinates 6, 6; 4, 4; 2, 2, &c. and through the points

1 5, 3, 1, k, &c. trace the curve n k m, then apd and nhm

! will be the two ribs for the centre: this being done, place

| | the ribs perpendicular over the lines a d and n m, and cover| them, as centres usually are, and then, applying the edge of

j a plumb-rule to the divers parts of the inside and outside

I of the window’s bottom, the top of the rule will give the

| several points at which the inside and outside of the covering

i is to be cut off, so as to stand exactly over the inside and

! outside of the building, and then the centre will be completed

' as required.”

I It is hardly possible to conceive anything so unscientificj as this description. In describing and forming the centre

i for the head of the required aperture, he is accurate ; but

; when we are told to apply “ the edge of a plumb-rule to the

divers parts of the inside and outside of the window’s bottom,”and that “ the top of the rule will give the several points atwhich the inside and outside of the covering are to be cutoff, so as to stand exactly over the inside and outside of thebuilding; and” that '• the centre will be completed as required,”he is altogether intolerable; for besides being tedious to an! extreme, it is no more than every mechanic could have easilyconceived. In forming the centre, it would be better to formthe inside curve with a trammel, which would obviate the

tedious work of dividing the base of each curve into equalparts, as well as the transferring of the ordinates of the semi"circle to those of the ellipses, and then, at last, either tracingthe elliptic curve by hand, or bending a thin slip of woodround pins or nails stuck in the points. At all events, evenin the operation of tracing, the dividing of the bases intoequal parts is a very bad practice, as it always leaves so larg eand so quick a portion of the curve at each extremity to beguessed at; but here it is admissible, on account of the fol'lowing diagrams connected therewith.

“ To divide the courses in the arch of this window. —On 11fiat panel, &c. draw a line, as be, in Figure 7, make a f°.equal to the curve a c d, also make a b and o e each equ® 1to the intended height of the brick arch. Make f ft lDFigure 7, equal to f p, in Figure 6; also make a b and o e <in Figure 7, each equal to b a, in Figure 6; then the poin tsb and e will be the extremes of the arch, Make p lX \Figure 7, equal to b a, the given height of the arch, a® 0through the points ire and a p o describe two semi-ellip 3 ®®!which divide into courses as before taught, and which wmbe the face of the arch required.”

This operation produces nothing, as he does not showapplication to practice, in the formation of the stones or brickto their proper shapes. . ,

“ To find the angles or bevels of the under part of eaC ,course. —Continue the splay backs of the window m d ® n ,n a until they meet in f. On f, with the radius f n » n .f a, describe the arches n y v and a f s, making n y v e<l u ‘ lto the girt of the arch n k m. Make n 6, n 4, n 2, n y> ^on the arch n y v, equal to n 6, n 4, n 2, n y, &c. on jycurve n k m, and draw the lines 6 f, 4 F, 2 F, y D f 'Make the ordinates 6, 5 ; 4, 3 ; 2, 1 ; y x, &c. on the6 f, 4 F, &c. equal to the ordinates 5, 6 ; 3, 4 ; 1, 2 ; h jon the line n m and through the points 5, 3, 1, x, &C-the curve v x n. In the same manner transfer the ordin®*

5, 6 ; 3, 4 ; 1, 2 ; cf &c. on the line a d to the arch S Jas from 5 to 6, from 4 to 3, &e. and trace the curve s c ’.

d o*

and then will the figure nxv sc a be the soffito of the winu ” 0 flaid out, and which being divided into the same numberequal parts, as the under part of the arch a p o, Fig uT ^ (0and lines drawn to the centre f, as is done in Figurethe centre a, by the lines 2, 2, 2, &c. those lines will gthe bevel of every course in soffito, as required.” p a

Here is an attempt to find the lining or envelope jcuneoid or cono-cuneus, in a circular wall, for the sou 1 ^the stones or bricks; and had he succeeded, his endea'' ^would have been so far right: but the method whic 1 .

follows has no relation to the construction of the citself, and is therefore extremely erroneous. Nor can .same method be applied to the covering of a cone, ththe affinity or relation is much nearer in the latter solid iin the former, and, consequently, the envelope herewould cover a cone more nearly than the surface of a cuBut, indeed, though very near approaches have been ^to the cuneoidal surface, its determinate figure hasbeen exactly shown on a plane : however, the S e0 '” e - fse lf,

the P

construction may be laid out on the surface of the suu" j a niand all curves, corresponding to given ones on the Vfound with the utmost accuracy. The other parts of FV - t jj-are not described in the text, but seem to contain li ne ®,.out meaning. The following is all that is said of TV1, 2, 4, and 5 : . • • 1 a^ 1 ’’

Figure 5 “ is another example of a semi-ellipt' cal piewhose front is Figure 2. Also Figure 4, is a third ex . j,of a scheme arch, whose front is Figure 8. And TV ^ fl r 8is a fourth example of a straight arch, which, in gener >performed by the aforesaid rule.”

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