668 
OPTICS. 
and from the points i and 2, draw the ftraight lines Li 
and Ms. Laftly, fince HI is the altitude of the intended 
cube in a, Li in c and b, M2 in d, draw from the point 
a the ftraight line fa perpendicular to aE, and from the 
points b and c, bg and ce, perpendicular to be 1 ; and, aide 
being according to rule, make a/=HI, bg=ec— Li, and 
hd=z M2. Then, if the points g, h, e, f, be joined, the 
whole cube will be in perfpeftive. 
Prob. XIV. To put a fquare pyramid in perfpeftive, as 
ftanding upright on its bafe, and viewed obliquely. 
Let AD, fig. 19, be the breadth of either of the four 
fides of the pyramid ATCD at its bafe ABCD ; and MT 
its perpendicular height. Let O be the place oftheobfer- 
ver, Shis point of fight, SE his horizon, parallel to AD and 
perpendicular to OS 5 and let the proper point of diftance 
be taken in SE produced toward the left hand, as far from 
S as O is from S. Draw AS and DS to the point of fight, 
and DL to the point of diftance, interfering AS in the 
point B. Then, from B draw BC parallel to AD; and 
ABCD fliall be the perfpeftive fquare bafe of the pyra¬ 
mid. Draw the diagonal AC, interfering the other 
diagonal BD at M, and this point of interferion fliall 
be the centre of the fquare bafe. Draw MT perpendicu¬ 
lar to AD, and of a length equal to the intended height 
of the pyramid : then draw the ftraight outlines AT, CT, 
and DT; and the outlines of the pyramid (as viewed 
from O) will be finilhed; which being done, the whole 
may be fo Ihaded as to give it the appearance of a folid 
body. 
It the obferver had flood at o, he could have only feen 
the fide ATD of the pyramid ; and two is the greateft 
number of fides that he could fee from any other place 
of the ground. But if he were at any height above the 
pyramid, and had his eye direriy over its top, it would 
then appear as in fig. 20. and he would fee all its four fides 
E, F, G, H, with its top t juft over the centre of its fquare 
bafe ABCD ; which would be a true geometrical (and not 
a perfpeftive) fquare. 
Prob. XV. To put two equal fquares in perfpeftive, one 
of which lhall be direftly over the other, at any given 
diftance from it, and both of them parallel to the plane 
of the horizon. 
Let ABCD, fig. 21, be a perfpeftive fquare on a hori¬ 
zontal plane, drawn according to the foregoing rules, _S 
being the point of fight, SP the horizon (parallel to AD), 
and P the point of diftance. 
Suppofe AD, the breadth of this fquare, to be three 
feet; and that it is required to place juft fuch another 
fquare EFGH direftly above it, parallel to it, and tw'o 
feet from it. Make AE and DH perpendicular to AD, 
and two-thirds of its length : draw EH, which will be 
equal and parallel to AD ; then draw ES and HS to the 
point of fight S, and EP to the point of diftance P, inter¬ 
fering HS in the point G. This done, draw FG parallel 
to EH ; and you will have two perfpeftive fquares, ABCD 
and EFGH, equal and parallel to one another, the latter 
direftly above the former, and two feet diftant from it; 
as was required. 
By this method (helves may be drawn parallel to one 
another, at any diftance from each other, in proportion 
to their length. 
Prob. XVI. To put a truncated pyramid in perfpeftive. 
Let the pyramid to be put in perfpeftive be quinquan- 
gular. If from, each angle of the furface whence the top 
is cut off, a perpendicular be fuppofed to fall upon the 
bafe, thefe perpendiculars will mark the bounding points 
of a pentagon, of which the fides will be parallel to the 
fides of the bafe of the pyramid within which it is in- 
feribed. Join thefe points, and the interior pentagon 
will be formed with its longeft fide parallel to the longert 
fide of the bafe of the pyramid. From the ground-line 
EH, fig. 22, raile the perpendicular IH, and make it equal 
to the altitude of the intended pyramid. To any point 
V draw the ftraight lines IV and HV, and, by a procefs 
fimilar to that in Prob. XIII. determine the feenogra- 
phical altitudes a, b, c, d, e. Conneft the upper points 
/, g'j h by ftraight lines; and draw Ik, fm, gn , and 
the perfpeftive of the truncated pyramid will be com¬ 
pleted. 
Cor. If in a geometrical plane two concentric circles be 
deferibed, a truncated cone may be put in perfpeftive in 
the fame manner as a truncated pyramid. 
Proe. XVII. To put in perfpeftive a hollow prifm lying 
on one of its fides. 
Let ABDEC, fig. 23, be a feftion of fuch a prifm. 
Draw HI parallel to AB, and diftant from it the breadth 
of the fide on which the prifm refts; and from each angle 
internal and external of the prifm let fall perpendiculars 
to HI. The parallelogram will be thus divided by the 
ichnographical procefs below the ground-line, fo as that 
the fide AB of the real prifm will be parallel to the cor- 
refponding fide of the fcenographic view of it. 
To determine the altitude of the internal and external 
angles : From H, fig. 24., raife HI perpendicular to the 
ground-line, and on it mark off the true altitudes Hi, 
H2, H3, H4, and H5. Then, if from any point V in the 
horizon be drawn the ftraight lines VH, Vi, V2, V3, V4, 
V5, or VI; by a procefs fimilar to that of the preceding 
Problem, will be determined the height of the internal 
angles, viz. 1 —aa, 2 ~bb, \—dd; and, of the external 
angles, 3 z=cc, and 5=ee; and, when thefe angles are 
formed and put in their proper places, the feenograph of 
the prifm is complete. 
Prob. XVIII. To put a fquare table in perfpeftive, (land¬ 
ing on four upright fquare legs of any given length 
with refpeft to the breadth of the table. 
Let ABCD, fig. 21, be the fquare part of the floor on 
which the table is to (land, and EFGH the furface of the 
fquare table, parallel to the floor. Suppofe the table to 
be three feet in breadth, and its height from the floor to 
be two feet; then two-thirds of AD or EH will be the 
length of the legs i and It ; the other two (l and m) being 
of the fame length in perfpeftive. 
Having drawn the two equal and parallel fquares 
ABCD and EFGH, as (hown in Prob. XV. let the legs be 
fquare in form, and fixed into the table at a diftance from 
its edges equal to their thicknefs. Take A a and Dd 
equal to the intended thicknefs of the legs, and ab and 
dc alfo equal thereto. Draw the diagonals AC and BD, 
and draw ftraight lines from the points a, b, c, d, towards 
the point of fight S, and terminating at the fide BC. 
Then, through the points where thefe lines cut the dia¬ 
gonals, draw the ftraight lines n and o, p and q, parallel 
to AD ; and you will have formed four perfpeftive fquares 
(like ABCD in fig. 19.) for the bales of the four legs of 
the table; and then it is eafy to draw the four upright 
legs by parallel lines, all perpendicular to AD, and to 
(hade them as in the figure. 
To reprefent the intended thicknefs of the table-board, 
draw eh. parallel to EH, and HG toward the point of fight 
S : then lhade the fpaces between thefe lines, and the 
perfpeftive figure of the table will be finilhed. 
Anamorphosis, or Re-formation of diftorted Images. 
By the means of what.is called anamorpliofis, piftures 
that are fo milhapen as to exhibit no regular appearance 
of any thing to the naked eye, fliall, when viewed by re- 
fleftion, prefent a regular and beautiful image. The in¬ 
ventor of this ingenious device is not known. Simon 
Stevinus, who was the firft that wrote upon it, does not 
inform us from whom he learned it. The principles of 
it are laid down by S. Vauzelard in his Perfpe&ive Conique 
et Cylindrique; and Gafpar Schott profefles to copy Ma¬ 
rius Bettinus in his defeription of this piece of artificial 
magic. It will be fufficient for our purpofe to copy one 
