356 LOWELL—THE CARTOUCHES OF MARS. LDee. 4, 
however, their full import two corrections had in rigor to be 
taken into account: one for the varying distance of the planet and 
the other for the varying quality of the seeing. At the several 
presentations the planet was not at the same distance from the 
Earth. Now distance affects the visibility of a marking by altering 
its size. If the markings be large, their apparent size decreases as 
the square of the distance. If, as in the case with the ‘‘ canals,”’ 
they have length without width, we may take them as of one 
dimension. For beyond a certain length increase of that quantity 
does not seriously affect the visibility. Their width, however, 
although unrecognizable as such, improves their chance of being 
seen in the direct ratio of the planet’s approach. 
Now if we take the chance that a canal of twice the width of a 
given one is twice as likely to be made out, we may regard it as the 
inverse of the relative chance of commission of twice a given error 
of observation. We may then use the areas bounded by the curve 
of probability, with the width of the canals taken for abscissz, 
respectively as the measures of the likelihood of detection in the 
two cases, since these areas include all the chances of seeing a canal 
of the given width. By taking the area from the central ordinate 
of the curve out to where that area shall equal the percentage of 
visibility shown at a given distance, then multiplying the ordinate 
there found by the inverse ratio of the given distance of the planet 
at the time to some fixed distance taken as standard, and then find- 
ing the area corresponding to this last ordinate, we shall get the 
percentage at the standard distance. It will be noted that on this 
principle, as the planet approaches the Earth the percentage of visi- 
bility increases gradually to unity, that is certainty of detection if 
the object exist at the time, since the area enclosed by the curve of 
probability approaches unity as the abscissa is indefinitely increased. 
For standard distance I took that of the planet’s nearest approach 
to us during the opposition, when its disk subtended 14”.6 of arc. 
On this principle have been computed the corrections for distance. 
The correction for the seeing was got in the following way. The 
seeing at the time of each drawing was entered in the course of obser- 
vation by the side of the drawing, together with all the other 
marginal notes. By taking the mean of these values for all the 
drawings which entered into the determination of the percentage 
visibility of a given canal at a given presentation, we get the mean 
seeing under which it was observed. The correction needed in 
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