'583 OPT 
The proper method of drawing the appearance of two 
rows of trees that (hall appear parallel to the eye, is a pro¬ 
blem which has exercifed the ingenuity of feveral philo- 
fophers and mathematicians. That the apparent magni¬ 
tude of objefts decreafes with the angle under which they 
are feen, has always been acknowledged. It is alfo acknow¬ 
ledged, that it is only by cuftom and experience that we 
learn to form a judgment both of magnitudes and dif- 
tances. But in the application of thefe maxims to the 
above-mentioned problems, all perfons, before M. Bou- 
guer, made ufe of the real diftance inftead of the apparent 
one, by which only the mind can form its judgment. And 
it is manifeft, that, if any circumltances contribute to make 
the diftance appear otherwife than it is in reality, the ap¬ 
parent magnitude of the objeft will be affefted by it; for 
the fame reafon that, if the magnitude be mifapprehended, 
the idea of the diftance will vary. For want of attending 
to this diftinflion, Tacquet pretended to demonftrate, 
that nothing can give the idea of two parallel lines (rows 
of trees for inftance) to an eye fituated at one of their ex¬ 
tremities, but two hyperbolical curves turned the con¬ 
trary way; and M. Varignon maintained, that, in order 
to make a vifta appear of the fame width, it mult be made 
narrower, inftead of wider, as it recedes from the eye. 
M. Bouguer obferves, that very great diftances, and 
thofe that are Considerably lefs than they, make nearly the 
fame impreflion upon the eye. We therefore always ima¬ 
gine great diftances to be lefs than they are ; and for this 
reafon the ground-plan of a long vifta always appears to 
rife. The vifual rays come in a determinate direffion ; 
but, as we imagine that they terminate fooner than they 
do, we neceflarily conceive that the place from which 
they iffue is elevated. Every large plane, therefore, as AB, 
(fig. 13.) viewed by an eye at O, will feem to lie in fuch a 
dire&ion as Ab; and confequently lines, in order to ap¬ 
pear truly parallel on the plane AB, muft be drawn fo as 
that they would appear parallel on the plane Ad, and be 
from thence projefted to the plane AB. To determine 
the inclination of the apparent ground-plan Ad to the 
true ground-plan AB, our ingenious author directs us to 
draw upon a piece of level ground two ftraight lines of a 
Sufficient length (for which purpofe lines faftened to fmall 
j’ticks are very convenient), making an angle of 3 or 4 
degrees with one another. Then a perfon, placing him- 
ielf within the angle with his back towards the angular 
point, muff walk backwards and forwards, till he can 
fancy the lines to be parallel. In this fttuation, a line 
drawn from the point of the angle through the place of 
his eye will contain the fame angle with the true ground- 
plan which this does with the apparent one. 
M. Bouguer then Shows other more geometrical me¬ 
thods of determining this inclination; and fays, that, by 
thefe means, he has often found it to be 4 or 5 degrees, 
though fometimec only 2 or 2J degrees. The determina¬ 
tion of this angle, he obferves, is variable; depending 
upon the manner in which the ground is illuminated, 
and the intenfity of the light. The colour of the foil is 
alfo not without its influence, as well as the particular 
conformation of the eye, by which it is more or lefs af- 
fefted by the fame degree of light, and alfo the part of the 
eye on which the objeft is painted. When, by a flight 
motion of his head, he contrived, that certain parts of 
the foil, the image of which fell towards the bottom of 
liis eye, fliould fall towards the top of the retina, he al¬ 
ways thought that this apparent inclination became a 
-little greater. 
But what is very remarkable, is, that if we look to¬ 
wards a riling ground, the difference between the appa¬ 
rent ground-plan and the true one will be much more 
sonfiderable, fo that they will fometimes make an angle 
of 25 or 30 degrees. Of this he had made frequent ob- 
fervations. Mountains, he fays, begin to be inacceftible 
when their Tides make an angle from 35 to 37 degrees 
with the horizon, as then it is not poflible to climb them 
ibut by means of ftones 0c Shrubs, to Serve as fteps to fix 
I c s. 
the feet on. In thefe cafes, both he and his companions 
always agreed that the apparent inclination of the fide of 
the mountain was 60 or 70 degrees. 
Thefe deceptions are reprefented in fig. 14. in which, 
when the ground-plan AM, or An, is much inclined, the 
apparent ground-plan Am, or An, makes a very large an¬ 
gle with it. On the contrary, if the ground dips'below 
the level, the inclination of the apparent to the true 
ground-plan diminilhes, till, at a certain degree of the 
flope, it becomes nothing at all ; the two plans AP and 
A p being the fame, fo that parallel lines drawn upon them 
would always appear fo. If the inclination below the ho¬ 
rizon is carried beyond the fituation AP, the error will 
increafe ; and, what is very remarkable, it will be on the 
contrary fide; the apparent plan A r being always below 
the true plan AR; fo that, if a perfon would draw upon 
the plan AR lines that fhall appear parallel to the eye, 
they muft be drawn converging, and not diverging, as is 
ufual on the level ground ; hecaufe they muft be the pro¬ 
jections of two lines imagined to be parallel on the plan 
A r, which is more inclined to the horizon than AR. 
Thefe remarks, he obferves, are applicable to different 
planes expofed to the eye at the fame time. For, if BH, 
fig. 13. be the front of a building, at the diftance of AB 
from the eye, it will be reduced in appearance to the dif¬ 
tance A b; and the front of the building will be b/i; ra¬ 
ther inclined, towards the fpeftator, unlels the diftance be 
inconfiderable. 
After making a great number of obfervations upon this 
fubjedt, our author concludes, that, when a man Hands 
upon a level plane, it does not feem to rife fenfibly but 
at fome diftance from him. The apparent plane, there¬ 
fore, has a curvature in it, at that diftance, the form of 
which is not very eafy to determine; fo that a man Stand¬ 
ing upon a level plane of infinite extent, will imagine 
that he Hands in the centre of a bafon. This is alfo in 
fome meafure the cafe with a perfon Handing upon the 
level of the fea. 
He concludes with obferving, that there is no difficulty 
in drawing lines according to thefe rules, fo as to have 
any given eft’eft upon the eye, except when fome parts of 
theprofpedt are very near the fpedtator, and others very 
diftant from him; becaufe, in this cafe, regard muft be had 
to the conical or conoidal figure of a furface. A right 
line palling at a fmall diftance from the obferver, and be¬ 
low the level of his eye, in that caSealmoSl always appears 
fenfibly curved at a certain diftance from the eye; and al¬ 
most all figures, in this cafe, are Subject to fome compli¬ 
cated optical alteration, to which the rules of perspective 
have not as yet been extended. If a circle be drawn near 
our feet, and within that part of the ground which ap¬ 
pears level to us, it will always appear to be a circle ; and, 
at a very considerable diftance, it will appear an elliple ; 
but, between thefe two fituations, it will not appear to be 
either the one or the ether, but will be like one of tbofe 
ovals of Des Cartes, which is more curved on one of its 
fides than the other. On thefe principles, a parterre, 
which appears diftorted when it is leen in a low fituation, 
appears perfectly regular when it is viewed from a bal¬ 
cony, or any other eminence. Still, however, the appa¬ 
rent irregularity takes place at a greater diftance, while 
the part that is near the fpedtator is exempt from it. If 
AB, fig. 16. be the ground-plan, and Aa be a perpendi¬ 
cular under the eye, the higher it is fituated, at O, to the 
greater diftance will T, the place at which the plane be¬ 
gins to have an apparent afeent along T b, be removed. 
M. Le Cat well explains a remarkable deception, by 
which a perfon Shall imagine an object to be on the oppo¬ 
site fide of aboard, when it is not So, and alfo inverted and 
magnified. It is illustrated by fig. 17. in which D repre- 
fents the eye, and CB a large black board, pierced wuth a 
fmall hole. E is a large white board, placed beyond it, 
and Strongly illuminated ; and d a pin, or other Small ob¬ 
ject, held betwixt the eye and the firft board. In theS’ecir- 
cumftances, the pin Shall be imagined to be at F, on the 
2 other 
